Beginning of Section Conj_mul_SNo_assoc_lem1__26__21

(*** $I sig/Nov2021ConjPreamble.mgs ***)

L4

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__26__21 TMXrPhiBRCMz9pTqLHTdFycxb5uuBkvWMqy bounty of about 25 bars ***)

L5

Variable x : set

L6

Variable y : set

L7

Variable z : set

L8

Variable w : set

L9

Variable u : set

L10

Variable v : set

L11

Variable x2 : set

L12

Variable y2 : set

L13

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L14

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L15

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L16

Hypothesis H3 : SNo x

L17

Hypothesis H4 : SNo y

L18

Hypothesis H5 : SNo z

L19

Hypothesis H6 : SNo (g x y)

L20

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L21

Hypothesis H8 : u ∈ SNoR x

L22

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L23

Hypothesis H10 : SNo u

L24

Hypothesis H11 : x < u

L25

Hypothesis H12 : SNo v

L26

Hypothesis H13 : x2 ∈ SNoR y

L27

Hypothesis H14 : y2 ∈ SNoL z

L28

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L29

Hypothesis H16 : SNo x2

L30

Hypothesis H17 : y < x2

L31

Hypothesis H18 : SNo y2

L32

Hypothesis H19 : y2 < z

L33

Hypothesis H20 : SNo (g u y)

L34

Hypothesis H22 : SNo (g x x2)

L35

Hypothesis H23 : SNo (g x2 z + g y y2)

L36

Hypothesis H24 : SNo (v + g x2 y2)

L37

Hypothesis H25 : SNo (g x (v + g x2 y2))

L38

Hypothesis H26 : SNo (g u (v + g x2 y2))

L39

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L40

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L41

Hypothesis H29 : SNo (g x y + g u x2)

L42

Theorem. (Conj_mul_SNo_assoc_lem1__26__21)

SNo (g (g x y + g u x2) y2) → w < g (g x y) z

In Proofgold the corresponding term root is 5821fa... and proposition id is bb8dfc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__27__27

L48

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__27__27 TMFNE4P28v5aLvucyhNssEyDR7eCevs2fbv bounty of about 25 bars ***)

L49

Variable x : set

L50

Variable y : set

L51

Variable z : set

L52

Variable w : set

L53

Variable u : set

L54

Variable v : set

L55

Variable x2 : set

L56

Variable y2 : set

L57

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L58

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L59

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L60

Hypothesis H3 : SNo x

L61

Hypothesis H4 : SNo y

L62

Hypothesis H5 : SNo z

L63

Hypothesis H6 : SNo (g x y)

L64

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L65

Hypothesis H8 : u ∈ SNoR x

L66

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L67

Hypothesis H10 : SNo u

L68

Hypothesis H11 : x < u

L69

Hypothesis H12 : SNo v

L70

Hypothesis H13 : x2 ∈ SNoR y

L71

Hypothesis H14 : y2 ∈ SNoL z

L72

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L73

Hypothesis H16 : SNo x2

L74

Hypothesis H17 : y < x2

L75

Hypothesis H18 : SNo y2

L76

Hypothesis H19 : y2 < z

L77

Hypothesis H20 : SNo (g u y)

L78

Hypothesis H21 : SNo (g u x2)

L79

Hypothesis H22 : SNo (g x x2)

L80

Hypothesis H23 : SNo (g x2 z + g y y2)

L81

Hypothesis H24 : SNo (v + g x2 y2)

L82

Hypothesis H25 : SNo (g x (v + g x2 y2))

L83

Hypothesis H26 : SNo (g u (v + g x2 y2))

L84

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L85

Hypothesis H29 : SNo (g u y + g x x2)

L86

Hypothesis H30 : SNo (g x y + g u x2)

L87

Theorem. (Conj_mul_SNo_assoc_lem1__27__27)

SNo (g (g u y + g x x2) y2) → w < g (g x y) z

In Proofgold the corresponding term root is 48f51a... and proposition id is e7121f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__27__30

L93

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__27__30 TMc2qDZ6zjzAU41rNAxu6v3fWbTz5vShdYw bounty of about 25 bars ***)

L94

Variable x : set

L95

Variable y : set

L96

Variable z : set

L97

Variable w : set

L98

Variable u : set

L99

Variable v : set

L100

Variable x2 : set

L101

Variable y2 : set

L102

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L103

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L104

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L105

Hypothesis H3 : SNo x

L106

Hypothesis H4 : SNo y

L107

Hypothesis H5 : SNo z

L108

Hypothesis H6 : SNo (g x y)

L109

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L110

Hypothesis H8 : u ∈ SNoR x

L111

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L112

Hypothesis H10 : SNo u

L113

Hypothesis H11 : x < u

L114

Hypothesis H12 : SNo v

L115

Hypothesis H13 : x2 ∈ SNoR y

L116

Hypothesis H14 : y2 ∈ SNoL z

L117

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L118

Hypothesis H16 : SNo x2

L119

Hypothesis H17 : y < x2

L120

Hypothesis H18 : SNo y2

L121

Hypothesis H19 : y2 < z

L122

Hypothesis H20 : SNo (g u y)

L123

Hypothesis H21 : SNo (g u x2)

L124

Hypothesis H22 : SNo (g x x2)

L125

Hypothesis H23 : SNo (g x2 z + g y y2)

L126

Hypothesis H24 : SNo (v + g x2 y2)

L127

Hypothesis H25 : SNo (g x (v + g x2 y2))

L128

Hypothesis H26 : SNo (g u (v + g x2 y2))

L129

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L130

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L131

Hypothesis H29 : SNo (g u y + g x x2)

L132

Theorem. (Conj_mul_SNo_assoc_lem1__27__30)

SNo (g (g u y + g x x2) y2) → w < g (g x y) z

In Proofgold the corresponding term root is c6f350... and proposition id is abd44a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__28__4

L138

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__28__4 TMMd41ujZEZfsW5b4MA3xE9JD6akW8DZWAb bounty of about 25 bars ***)

L139

Variable x : set

L140

Variable y : set

L141

Variable z : set

L142

Variable w : set

L143

Variable u : set

L144

Variable v : set

L145

Variable x2 : set

L146

Variable y2 : set

L147

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L148

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L149

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L150

Hypothesis H3 : SNo x

L151

Hypothesis H5 : SNo z

L152

Hypothesis H6 : SNo (g x y)

L153

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L154

Hypothesis H8 : u ∈ SNoR x

L155

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L156

Hypothesis H10 : SNo u

L157

Hypothesis H11 : x < u

L158

Hypothesis H12 : SNo v

L159

Hypothesis H13 : x2 ∈ SNoR y

L160

Hypothesis H14 : y2 ∈ SNoL z

L161

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L162

Hypothesis H16 : SNo x2

L163

Hypothesis H17 : y < x2

L164

Hypothesis H18 : SNo y2

L165

Hypothesis H19 : y2 < z

L166

Hypothesis H20 : SNo (g u y)

L167

Hypothesis H21 : SNo (g u x2)

L168

Hypothesis H22 : SNo (g x x2)

L169

Hypothesis H23 : SNo (g x2 z + g y y2)

L170

Hypothesis H24 : SNo (v + g x2 y2)

L171

Hypothesis H25 : SNo (g x (v + g x2 y2))

L172

Hypothesis H26 : SNo (g u (v + g x2 y2))

L173

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L174

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L175

Hypothesis H29 : SNo (g u y + g x x2)

L176

Hypothesis H30 : SNo (g x y + g u x2)

L177

Theorem. (Conj_mul_SNo_assoc_lem1__28__4)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is 7f288d... and proposition id is c230ba...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__28__7

L183

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__28__7 TMN5iEjnyUjoaC7uGHCujZGWddhzeiEw83b bounty of about 25 bars ***)

L184

Variable x : set

L185

Variable y : set

L186

Variable z : set

L187

Variable w : set

L188

Variable u : set

L189

Variable v : set

L190

Variable x2 : set

L191

Variable y2 : set

L192

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L193

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L194

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L195

Hypothesis H3 : SNo x

L196

Hypothesis H4 : SNo y

L197

Hypothesis H5 : SNo z

L198

Hypothesis H6 : SNo (g x y)

L199

Hypothesis H8 : u ∈ SNoR x

L200

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L201

Hypothesis H10 : SNo u

L202

Hypothesis H11 : x < u

L203

Hypothesis H12 : SNo v

L204

Hypothesis H13 : x2 ∈ SNoR y

L205

Hypothesis H14 : y2 ∈ SNoL z

L206

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L207

Hypothesis H16 : SNo x2

L208

Hypothesis H17 : y < x2

L209

Hypothesis H18 : SNo y2

L210

Hypothesis H19 : y2 < z

L211

Hypothesis H20 : SNo (g u y)

L212

Hypothesis H21 : SNo (g u x2)

L213

Hypothesis H22 : SNo (g x x2)

L214

Hypothesis H23 : SNo (g x2 z + g y y2)

L215

Hypothesis H24 : SNo (v + g x2 y2)

L216

Hypothesis H25 : SNo (g x (v + g x2 y2))

L217

Hypothesis H26 : SNo (g u (v + g x2 y2))

L218

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L219

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L220

Hypothesis H29 : SNo (g u y + g x x2)

L221

Hypothesis H30 : SNo (g x y + g u x2)

L222

Theorem. (Conj_mul_SNo_assoc_lem1__28__7)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is bfaa45... and proposition id is a597a0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__28__8

L228

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__28__8 TMYvEA2jPfddvxXzaWsFTu95aLfxfu9HWBP bounty of about 25 bars ***)

L229

Variable x : set

L230

Variable y : set

L231

Variable z : set

L232

Variable w : set

L233

Variable u : set

L234

Variable v : set

L235

Variable x2 : set

L236

Variable y2 : set

L237

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L238

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L239

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L240

Hypothesis H3 : SNo x

L241

Hypothesis H4 : SNo y

L242

Hypothesis H5 : SNo z

L243

Hypothesis H6 : SNo (g x y)

L244

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L245

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L246

Hypothesis H10 : SNo u

L247

Hypothesis H11 : x < u

L248

Hypothesis H12 : SNo v

L249

Hypothesis H13 : x2 ∈ SNoR y

L250

Hypothesis H14 : y2 ∈ SNoL z

L251

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L252

Hypothesis H16 : SNo x2

L253

Hypothesis H17 : y < x2

L254

Hypothesis H18 : SNo y2

L255

Hypothesis H19 : y2 < z

L256

Hypothesis H20 : SNo (g u y)

L257

Hypothesis H21 : SNo (g u x2)

L258

Hypothesis H22 : SNo (g x x2)

L259

Hypothesis H23 : SNo (g x2 z + g y y2)

L260

Hypothesis H24 : SNo (v + g x2 y2)

L261

Hypothesis H25 : SNo (g x (v + g x2 y2))

L262

Hypothesis H26 : SNo (g u (v + g x2 y2))

L263

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L264

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L265

Hypothesis H29 : SNo (g u y + g x x2)

L266

Hypothesis H30 : SNo (g x y + g u x2)

L267

Theorem. (Conj_mul_SNo_assoc_lem1__28__8)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is 2234a6... and proposition id is 2c6af1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__28__22

L273

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__28__22 TMJLzJr7G23kN3w4TvLnjHtRYYWvaoic7rM bounty of about 25 bars ***)

L274

Variable x : set

L275

Variable y : set

L276

Variable z : set

L277

Variable w : set

L278

Variable u : set

L279

Variable v : set

L280

Variable x2 : set

L281

Variable y2 : set

L282

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L283

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L284

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L285

Hypothesis H3 : SNo x

L286

Hypothesis H4 : SNo y

L287

Hypothesis H5 : SNo z

L288

Hypothesis H6 : SNo (g x y)

L289

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L290

Hypothesis H8 : u ∈ SNoR x

L291

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L292

Hypothesis H10 : SNo u

L293

Hypothesis H11 : x < u

L294

Hypothesis H12 : SNo v

L295

Hypothesis H13 : x2 ∈ SNoR y

L296

Hypothesis H14 : y2 ∈ SNoL z

L297

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L298

Hypothesis H16 : SNo x2

L299

Hypothesis H17 : y < x2

L300

Hypothesis H18 : SNo y2

L301

Hypothesis H19 : y2 < z

L302

Hypothesis H20 : SNo (g u y)

L303

Hypothesis H21 : SNo (g u x2)

L304

Hypothesis H23 : SNo (g x2 z + g y y2)

L305

Hypothesis H24 : SNo (v + g x2 y2)

L306

Hypothesis H25 : SNo (g x (v + g x2 y2))

L307

Hypothesis H26 : SNo (g u (v + g x2 y2))

L308

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L309

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L310

Hypothesis H29 : SNo (g u y + g x x2)

L311

Hypothesis H30 : SNo (g x y + g u x2)

L312

Theorem. (Conj_mul_SNo_assoc_lem1__28__22)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is 0d0563... and proposition id is 541ead...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__28__27

L318

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__28__27 TMVAsQkVN74SrKoSGEubBkY4jhhnAQvay1G bounty of about 25 bars ***)

L319

Variable x : set

L320

Variable y : set

L321

Variable z : set

L322

Variable w : set

L323

Variable u : set

L324

Variable v : set

L325

Variable x2 : set

L326

Variable y2 : set

L327

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L328

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L329

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L330

Hypothesis H3 : SNo x

L331

Hypothesis H4 : SNo y

L332

Hypothesis H5 : SNo z

L333

Hypothesis H6 : SNo (g x y)

L334

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L335

Hypothesis H8 : u ∈ SNoR x

L336

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L337

Hypothesis H10 : SNo u

L338

Hypothesis H11 : x < u

L339

Hypothesis H12 : SNo v

L340

Hypothesis H13 : x2 ∈ SNoR y

L341

Hypothesis H14 : y2 ∈ SNoL z

L342

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L343

Hypothesis H16 : SNo x2

L344

Hypothesis H17 : y < x2

L345

Hypothesis H18 : SNo y2

L346

Hypothesis H19 : y2 < z

L347

Hypothesis H20 : SNo (g u y)

L348

Hypothesis H21 : SNo (g u x2)

L349

Hypothesis H22 : SNo (g x x2)

L350

Hypothesis H23 : SNo (g x2 z + g y y2)

L351

Hypothesis H24 : SNo (v + g x2 y2)

L352

Hypothesis H25 : SNo (g x (v + g x2 y2))

L353

Hypothesis H26 : SNo (g u (v + g x2 y2))

L354

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L355

Hypothesis H29 : SNo (g u y + g x x2)

L356

Hypothesis H30 : SNo (g x y + g u x2)

L357

Theorem. (Conj_mul_SNo_assoc_lem1__28__27)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is ef4b3a... and proposition id is a9a00f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__28__30

L363

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__28__30 TMHqLodnGtn2oxrfGJWwZqPvnMeB6JQGN3C bounty of about 25 bars ***)

L364

Variable x : set

L365

Variable y : set

L366

Variable z : set

L367

Variable w : set

L368

Variable u : set

L369

Variable v : set

L370

Variable x2 : set

L371

Variable y2 : set

L372

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L373

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L374

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L375

Hypothesis H3 : SNo x

L376

Hypothesis H4 : SNo y

L377

Hypothesis H5 : SNo z

L378

Hypothesis H6 : SNo (g x y)

L379

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L380

Hypothesis H8 : u ∈ SNoR x

L381

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L382

Hypothesis H10 : SNo u

L383

Hypothesis H11 : x < u

L384

Hypothesis H12 : SNo v

L385

Hypothesis H13 : x2 ∈ SNoR y

L386

Hypothesis H14 : y2 ∈ SNoL z

L387

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L388

Hypothesis H16 : SNo x2

L389

Hypothesis H17 : y < x2

L390

Hypothesis H18 : SNo y2

L391

Hypothesis H19 : y2 < z

L392

Hypothesis H20 : SNo (g u y)

L393

Hypothesis H21 : SNo (g u x2)

L394

Hypothesis H22 : SNo (g x x2)

L395

Hypothesis H23 : SNo (g x2 z + g y y2)

L396

Hypothesis H24 : SNo (v + g x2 y2)

L397

Hypothesis H25 : SNo (g x (v + g x2 y2))

L398

Hypothesis H26 : SNo (g u (v + g x2 y2))

L399

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L400

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L401

Hypothesis H29 : SNo (g u y + g x x2)

L402

Theorem. (Conj_mul_SNo_assoc_lem1__28__30)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is 36422c... and proposition id is 8b6e3b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__29__14

L408

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__29__14 TMcM49sfyRXJRnVHWkvE9fSGi7dEfsi1uVG bounty of about 25 bars ***)

L409

Variable x : set

L410

Variable y : set

L411

Variable z : set

L412

Variable w : set

L413

Variable u : set

L414

Variable v : set

L415

Variable x2 : set

L416

Variable y2 : set

L417

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L418

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L419

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L420

Hypothesis H3 : SNo x

L421

Hypothesis H4 : SNo y

L422

Hypothesis H5 : SNo z

L423

Hypothesis H6 : SNo (g x y)

L424

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L425

Hypothesis H8 : u ∈ SNoR x

L426

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L427

Hypothesis H10 : SNo u

L428

Hypothesis H11 : x < u

L429

Hypothesis H12 : SNo v

L430

Hypothesis H13 : x2 ∈ SNoR y

L431

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L432

Hypothesis H16 : SNo x2

L433

Hypothesis H17 : y < x2

L434

Hypothesis H18 : SNo y2

L435

Hypothesis H19 : y2 < z

L436

Hypothesis H20 : SNo (g u y)

L437

Hypothesis H21 : SNo (g u x2)

L438

Hypothesis H22 : SNo (g x x2)

L439

Hypothesis H23 : SNo (g x2 z + g y y2)

L440

Hypothesis H24 : SNo (v + g x2 y2)

L441

Hypothesis H25 : SNo (g x (v + g x2 y2))

L442

Hypothesis H26 : SNo (g u (v + g x2 y2))

L443

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L444

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L445

Hypothesis H29 : SNo (g u y + g x x2)

L446

Hypothesis H30 : SNo (g x y + g u x2)

L447

Theorem. (Conj_mul_SNo_assoc_lem1__29__14)

SNo (g (g u y + g x x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is fec08d... and proposition id is 7a69fe...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__30__0

L453

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__30__0 TMNGrNGThR2sRz5fAYgLgERdrGtb8zHm3Xb bounty of about 25 bars ***)

L454

Variable x : set

L455

Variable y : set

L456

Variable z : set

L457

Variable w : set

L458

Variable u : set

L459

Variable v : set

L460

Variable x2 : set

L461

Variable y2 : set

L462

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L463

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L464

Hypothesis H3 : SNo x

L465

Hypothesis H4 : SNo y

L466

Hypothesis H5 : SNo z

L467

Hypothesis H6 : SNo (g x y)

L468

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L469

Hypothesis H8 : u ∈ SNoR x

L470

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L471

Hypothesis H10 : SNo u

L472

Hypothesis H11 : x < u

L473

Hypothesis H12 : SNo v

L474

Hypothesis H13 : x2 ∈ SNoR y

L475

Hypothesis H14 : y2 ∈ SNoL z

L476

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L477

Hypothesis H16 : SNo x2

L478

Hypothesis H17 : y < x2

L479

Hypothesis H18 : SNo y2

L480

Hypothesis H19 : y2 < z

L481

Hypothesis H20 : SNo (g u y)

L482

Hypothesis H21 : SNo (g u x2)

L483

Hypothesis H22 : SNo (g x x2)

L484

Hypothesis H23 : SNo (g x2 z + g y y2)

L485

Hypothesis H24 : SNo (v + g x2 y2)

L486

Hypothesis H25 : SNo (g x (v + g x2 y2))

L487

Hypothesis H26 : SNo (g u (v + g x2 y2))

L488

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L489

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L490

Hypothesis H29 : SNo (g u y + g x x2)

L491

Theorem. (Conj_mul_SNo_assoc_lem1__30__0)

SNo (g x y + g u x2) → w < g (g x y) z

In Proofgold the corresponding term root is 045849... and proposition id is 7cf689...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__30__4

L497

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__30__4 TMUM1CXXxs8NNsypD9rA7cH7sSeXUtgLTqA bounty of about 25 bars ***)

L498

Variable x : set

L499

Variable y : set

L500

Variable z : set

L501

Variable w : set

L502

Variable u : set

L503

Variable v : set

L504

Variable x2 : set

L505

Variable y2 : set

L506

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L507

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L508

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L509

Hypothesis H3 : SNo x

L510

Hypothesis H5 : SNo z

L511

Hypothesis H6 : SNo (g x y)

L512

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L513

Hypothesis H8 : u ∈ SNoR x

L514

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L515

Hypothesis H10 : SNo u

L516

Hypothesis H11 : x < u

L517

Hypothesis H12 : SNo v

L518

Hypothesis H13 : x2 ∈ SNoR y

L519

Hypothesis H14 : y2 ∈ SNoL z

L520

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L521

Hypothesis H16 : SNo x2

L522

Hypothesis H17 : y < x2

L523

Hypothesis H18 : SNo y2

L524

Hypothesis H19 : y2 < z

L525

Hypothesis H20 : SNo (g u y)

L526

Hypothesis H21 : SNo (g u x2)

L527

Hypothesis H22 : SNo (g x x2)

L528

Hypothesis H23 : SNo (g x2 z + g y y2)

L529

Hypothesis H24 : SNo (v + g x2 y2)

L530

Hypothesis H25 : SNo (g x (v + g x2 y2))

L531

Hypothesis H26 : SNo (g u (v + g x2 y2))

L532

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L533

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L534

Hypothesis H29 : SNo (g u y + g x x2)

L535

Theorem. (Conj_mul_SNo_assoc_lem1__30__4)

SNo (g x y + g u x2) → w < g (g x y) z

In Proofgold the corresponding term root is 292c9c... and proposition id is 633da2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__31__23

L541

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__31__23 TMbXkZ56N1Wt4Li9th5L6HA7L52eCaYi12o bounty of about 25 bars ***)

L542

Variable x : set

L543

Variable y : set

L544

Variable z : set

L545

Variable w : set

L546

Variable u : set

L547

Variable v : set

L548

Variable x2 : set

L549

Variable y2 : set

L550

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L551

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L552

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L553

Hypothesis H3 : SNo x

L554

Hypothesis H4 : SNo y

L555

Hypothesis H5 : SNo z

L556

Hypothesis H6 : SNo (g x y)

L557

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L558

Hypothesis H8 : u ∈ SNoR x

L559

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L560

Hypothesis H10 : SNo u

L561

Hypothesis H11 : x < u

L562

Hypothesis H12 : SNo v

L563

Hypothesis H13 : x2 ∈ SNoR y

L564

Hypothesis H14 : y2 ∈ SNoL z

L565

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L566

Hypothesis H16 : SNo x2

L567

Hypothesis H17 : y < x2

L568

Hypothesis H18 : SNo y2

L569

Hypothesis H19 : y2 < z

L570

Hypothesis H20 : SNo (g u y)

L571

Hypothesis H21 : SNo (g u x2)

L572

Hypothesis H22 : SNo (g x x2)

L573

Hypothesis H24 : SNo (v + g x2 y2)

L574

Hypothesis H25 : SNo (g x (v + g x2 y2))

L575

Hypothesis H26 : SNo (g u (v + g x2 y2))

L576

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L577

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L578

Theorem. (Conj_mul_SNo_assoc_lem1__31__23)

SNo (g u y + g x x2) → w < g (g x y) z

In Proofgold the corresponding term root is d92b1d... and proposition id is 1531d7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__31__27

L584

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__31__27 TMHv5eedM5n1yicCqCV1zWEMX5woZ2TwW3i bounty of about 25 bars ***)

L585

Variable x : set

L586

Variable y : set

L587

Variable z : set

L588

Variable w : set

L589

Variable u : set

L590

Variable v : set

L591

Variable x2 : set

L592

Variable y2 : set

L593

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L594

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L595

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L596

Hypothesis H3 : SNo x

L597

Hypothesis H4 : SNo y

L598

Hypothesis H5 : SNo z

L599

Hypothesis H6 : SNo (g x y)

L600

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L601

Hypothesis H8 : u ∈ SNoR x

L602

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L603

Hypothesis H10 : SNo u

L604

Hypothesis H11 : x < u

L605

Hypothesis H12 : SNo v

L606

Hypothesis H13 : x2 ∈ SNoR y

L607

Hypothesis H14 : y2 ∈ SNoL z

L608

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L609

Hypothesis H16 : SNo x2

L610

Hypothesis H17 : y < x2

L611

Hypothesis H18 : SNo y2

L612

Hypothesis H19 : y2 < z

L613

Hypothesis H20 : SNo (g u y)

L614

Hypothesis H21 : SNo (g u x2)

L615

Hypothesis H22 : SNo (g x x2)

L616

Hypothesis H23 : SNo (g x2 z + g y y2)

L617

Hypothesis H24 : SNo (v + g x2 y2)

L618

Hypothesis H25 : SNo (g x (v + g x2 y2))

L619

Hypothesis H26 : SNo (g u (v + g x2 y2))

L620

Hypothesis H28 : SNo (g x (g x2 z + g y y2))

L621

Theorem. (Conj_mul_SNo_assoc_lem1__31__27)

SNo (g u y + g x x2) → w < g (g x y) z

In Proofgold the corresponding term root is 978465... and proposition id is 8ca155...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__33__4

L627

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__33__4 TMRofdExL1eRr8aSTg75nkNZbTjBQKyAq9t bounty of about 25 bars ***)

L628

Variable x : set

L629

Variable y : set

L630

Variable z : set

L631

Variable w : set

L632

Variable u : set

L633

Variable v : set

L634

Variable x2 : set

L635

Variable y2 : set

L636

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L637

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L638

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L639

Hypothesis H3 : SNo x

L640

Hypothesis H5 : SNo z

L641

Hypothesis H6 : SNo (g x y)

L642

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L643

Hypothesis H8 : u ∈ SNoR x

L644

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L645

Hypothesis H10 : SNo u

L646

Hypothesis H11 : x < u

L647

Hypothesis H12 : SNo v

L648

Hypothesis H13 : SNo (g u v)

L649

Hypothesis H14 : x2 ∈ SNoR y

L650

Hypothesis H15 : y2 ∈ SNoL z

L651

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L652

Hypothesis H17 : SNo x2

L653

Hypothesis H18 : y < x2

L654

Hypothesis H19 : SNo y2

L655

Hypothesis H20 : y2 < z

L656

Hypothesis H21 : SNo (g u y)

L657

Hypothesis H22 : SNo (g u x2)

L658

Hypothesis H23 : SNo (g x x2)

L659

Hypothesis H24 : SNo (g x2 z + g y y2)

L660

Hypothesis H25 : SNo (v + g x2 y2)

L661

Hypothesis H26 : SNo (g x (v + g x2 y2))

L662

Hypothesis H27 : SNo (g u (v + g x2 y2))

L663

Hypothesis H28 : SNo (g u (g x2 y2))

L664

Hypothesis H29 : SNo (g u (g x2 z + g y y2))

L665

Theorem. (Conj_mul_SNo_assoc_lem1__33__4)

SNo (g u v + g u (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is c4bfe6... and proposition id is 6d2973...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__33__24

L671

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__33__24 TMUEt815xSG2ZBSPJRPj4WijAreRFYxtxSd bounty of about 25 bars ***)

L672

Variable x : set

L673

Variable y : set

L674

Variable z : set

L675

Variable w : set

L676

Variable u : set

L677

Variable v : set

L678

Variable x2 : set

L679

Variable y2 : set

L680

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L681

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L682

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L683

Hypothesis H3 : SNo x

L684

Hypothesis H4 : SNo y

L685

Hypothesis H5 : SNo z

L686

Hypothesis H6 : SNo (g x y)

L687

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L688

Hypothesis H8 : u ∈ SNoR x

L689

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L690

Hypothesis H10 : SNo u

L691

Hypothesis H11 : x < u

L692

Hypothesis H12 : SNo v

L693

Hypothesis H13 : SNo (g u v)

L694

Hypothesis H14 : x2 ∈ SNoR y

L695

Hypothesis H15 : y2 ∈ SNoL z

L696

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L697

Hypothesis H17 : SNo x2

L698

Hypothesis H18 : y < x2

L699

Hypothesis H19 : SNo y2

L700

Hypothesis H20 : y2 < z

L701

Hypothesis H21 : SNo (g u y)

L702

Hypothesis H22 : SNo (g u x2)

L703

Hypothesis H23 : SNo (g x x2)

L704

Hypothesis H25 : SNo (v + g x2 y2)

L705

Hypothesis H26 : SNo (g x (v + g x2 y2))

L706

Hypothesis H27 : SNo (g u (v + g x2 y2))

L707

Hypothesis H28 : SNo (g u (g x2 y2))

L708

Hypothesis H29 : SNo (g u (g x2 z + g y y2))

L709

Theorem. (Conj_mul_SNo_assoc_lem1__33__24)

SNo (g u v + g u (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is f5211b... and proposition id is c257bf...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__34__8

L715

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__34__8 TMFVSBaPwqEiT2qMf9A4kHkr95DvR5nnR6p bounty of about 25 bars ***)

L716

Variable x : set

L717

Variable y : set

L718

Variable z : set

L719

Variable w : set

L720

Variable u : set

L721

Variable v : set

L722

Variable x2 : set

L723

Variable y2 : set

L724

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L725

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L726

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L727

Hypothesis H3 : SNo x

L728

Hypothesis H4 : SNo y

L729

Hypothesis H5 : SNo z

L730

Hypothesis H6 : SNo (g x y)

L731

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L732

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L733

Hypothesis H10 : SNo u

L734

Hypothesis H11 : x < u

L735

Hypothesis H12 : SNo v

L736

Hypothesis H13 : SNo (g u v)

L737

Hypothesis H14 : x2 ∈ SNoR y

L738

Hypothesis H15 : y2 ∈ SNoL z

L739

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L740

Hypothesis H17 : SNo x2

L741

Hypothesis H18 : y < x2

L742

Hypothesis H19 : SNo y2

L743

Hypothesis H20 : y2 < z

L744

Hypothesis H21 : SNo (g u y)

L745

Hypothesis H22 : SNo (g u x2)

L746

Hypothesis H23 : SNo (g x x2)

L747

Hypothesis H24 : SNo (g x2 z + g y y2)

L748

Hypothesis H25 : SNo (v + g x2 y2)

L749

Hypothesis H26 : SNo (g x (v + g x2 y2))

L750

Hypothesis H27 : SNo (g u (v + g x2 y2))

L751

Hypothesis H28 : SNo (g u (g x2 y2))

L752

Theorem. (Conj_mul_SNo_assoc_lem1__34__8)

SNo (g u (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is a4ad9e... and proposition id is 9eb95d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__34__9

L758

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__34__9 TMdg9DgeQbMXuEuKEArSbktjRv4vX5u9LSF bounty of about 25 bars ***)

L759

Variable x : set

L760

Variable y : set

L761

Variable z : set

L762

Variable w : set

L763

Variable u : set

L764

Variable v : set

L765

Variable x2 : set

L766

Variable y2 : set

L767

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L768

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L769

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L770

Hypothesis H3 : SNo x

L771

Hypothesis H4 : SNo y

L772

Hypothesis H5 : SNo z

L773

Hypothesis H6 : SNo (g x y)

L774

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L775

Hypothesis H8 : u ∈ SNoR x

L776

Hypothesis H10 : SNo u

L777

Hypothesis H11 : x < u

L778

Hypothesis H12 : SNo v

L779

Hypothesis H13 : SNo (g u v)

L780

Hypothesis H14 : x2 ∈ SNoR y

L781

Hypothesis H15 : y2 ∈ SNoL z

L782

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L783

Hypothesis H17 : SNo x2

L784

Hypothesis H18 : y < x2

L785

Hypothesis H19 : SNo y2

L786

Hypothesis H20 : y2 < z

L787

Hypothesis H21 : SNo (g u y)

L788

Hypothesis H22 : SNo (g u x2)

L789

Hypothesis H23 : SNo (g x x2)

L790

Hypothesis H24 : SNo (g x2 z + g y y2)

L791

Hypothesis H25 : SNo (v + g x2 y2)

L792

Hypothesis H26 : SNo (g x (v + g x2 y2))

L793

Hypothesis H27 : SNo (g u (v + g x2 y2))

L794

Hypothesis H28 : SNo (g u (g x2 y2))

L795

Theorem. (Conj_mul_SNo_assoc_lem1__34__9)

SNo (g u (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is ebdbd6... and proposition id is 45d75f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__35__4

L801

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__35__4 TMR3PewmRSCvFUF6dNJjVVpKMVgmGgUCjTv bounty of about 25 bars ***)

L802

Variable x : set

L803

Variable y : set

L804

Variable z : set

L805

Variable w : set

L806

Variable u : set

L807

Variable v : set

L808

Variable x2 : set

L809

Variable y2 : set

L810

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L811

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L812

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L813

Hypothesis H3 : SNo x

L814

Hypothesis H5 : SNo z

L815

Hypothesis H6 : SNo (g x y)

L816

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L817

Hypothesis H8 : u ∈ SNoR x

L818

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L819

Hypothesis H10 : SNo u

L820

Hypothesis H11 : x < u

L821

Hypothesis H12 : SNo v

L822

Hypothesis H13 : SNo (g u v)

L823

Hypothesis H14 : x2 ∈ SNoR y

L824

Hypothesis H15 : y2 ∈ SNoL z

L825

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L826

Hypothesis H17 : SNo x2

L827

Hypothesis H18 : y < x2

L828

Hypothesis H19 : SNo y2

L829

Hypothesis H20 : y2 < z

L830

Hypothesis H21 : SNo (g u y)

L831

Hypothesis H22 : SNo (g u x2)

L832

Hypothesis H23 : SNo (g x x2)

L833

Hypothesis H24 : SNo (g x2 z + g y y2)

L834

Hypothesis H25 : SNo (g x2 y2)

L835

Hypothesis H26 : SNo (v + g x2 y2)

L836

Hypothesis H27 : SNo (g x (v + g x2 y2))

L837

Hypothesis H28 : SNo (g u (v + g x2 y2))

L838

Theorem. (Conj_mul_SNo_assoc_lem1__35__4)

SNo (g u (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is fb9f35... and proposition id is 1fb085...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__36__3

L844

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__36__3 TMVHi8WazMbsjErVzhwp8Y9Vy3Wnxor3hqV bounty of about 25 bars ***)

L845

Variable x : set

L846

Variable y : set

L847

Variable z : set

L848

Variable w : set

L849

Variable u : set

L850

Variable v : set

L851

Variable x2 : set

L852

Variable y2 : set

L853

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L854

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L855

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L856

Hypothesis H4 : SNo y

L857

Hypothesis H5 : SNo z

L858

Hypothesis H6 : SNo (g x y)

L859

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L860

Hypothesis H8 : u ∈ SNoR x

L861

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L862

Hypothesis H10 : SNo u

L863

Hypothesis H11 : x < u

L864

Hypothesis H12 : SNo v

L865

Hypothesis H13 : SNo (g u v)

L866

Hypothesis H14 : x2 ∈ SNoR y

L867

Hypothesis H15 : y2 ∈ SNoL z

L868

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L869

Hypothesis H17 : SNo x2

L870

Hypothesis H18 : y < x2

L871

Hypothesis H19 : SNo y2

L872

Hypothesis H20 : y2 < z

L873

Hypothesis H21 : SNo (g u y)

L874

Hypothesis H22 : SNo (g u x2)

L875

Hypothesis H23 : SNo (g x x2)

L876

Hypothesis H24 : SNo (g x2 z + g y y2)

L877

Hypothesis H25 : SNo (g x2 y2)

L878

Hypothesis H26 : SNo (v + g x2 y2)

L879

Hypothesis H27 : SNo (g x (v + g x2 y2))

L880

Theorem. (Conj_mul_SNo_assoc_lem1__36__3)

SNo (g u (v + g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 0c2c0b... and proposition id is e08854...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__36__11

L886

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__36__11 TMFQaYLFPCE2DXPVx5wXCnQDLNuzjoLzvre bounty of about 25 bars ***)

L887

Variable x : set

L888

Variable y : set

L889

Variable z : set

L890

Variable w : set

L891

Variable u : set

L892

Variable v : set

L893

Variable x2 : set

L894

Variable y2 : set

L895

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L896

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L897

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L898

Hypothesis H3 : SNo x

L899

Hypothesis H4 : SNo y

L900

Hypothesis H5 : SNo z

L901

Hypothesis H6 : SNo (g x y)

L902

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L903

Hypothesis H8 : u ∈ SNoR x

L904

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L905

Hypothesis H10 : SNo u

L906

Hypothesis H12 : SNo v

L907

Hypothesis H13 : SNo (g u v)

L908

Hypothesis H14 : x2 ∈ SNoR y

L909

Hypothesis H15 : y2 ∈ SNoL z

L910

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L911

Hypothesis H17 : SNo x2

L912

Hypothesis H18 : y < x2

L913

Hypothesis H19 : SNo y2

L914

Hypothesis H20 : y2 < z

L915

Hypothesis H21 : SNo (g u y)

L916

Hypothesis H22 : SNo (g u x2)

L917

Hypothesis H23 : SNo (g x x2)

L918

Hypothesis H24 : SNo (g x2 z + g y y2)

L919

Hypothesis H25 : SNo (g x2 y2)

L920

Hypothesis H26 : SNo (v + g x2 y2)

L921

Hypothesis H27 : SNo (g x (v + g x2 y2))

L922

Theorem. (Conj_mul_SNo_assoc_lem1__36__11)

SNo (g u (v + g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is bdc4a6... and proposition id is f5488c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__36__23

L928

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__36__23 TMa9TLnUkpUaf3g9PCKPFgteDzSKJcJQJdb bounty of about 25 bars ***)

L929

Variable x : set

L930

Variable y : set

L931

Variable z : set

L932

Variable w : set

L933

Variable u : set

L934

Variable v : set

L935

Variable x2 : set

L936

Variable y2 : set

L937

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L938

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L939

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L940

Hypothesis H3 : SNo x

L941

Hypothesis H4 : SNo y

L942

Hypothesis H5 : SNo z

L943

Hypothesis H6 : SNo (g x y)

L944

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L945

Hypothesis H8 : u ∈ SNoR x

L946

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L947

Hypothesis H10 : SNo u

L948

Hypothesis H11 : x < u

L949

Hypothesis H12 : SNo v

L950

Hypothesis H13 : SNo (g u v)

L951

Hypothesis H14 : x2 ∈ SNoR y

L952

Hypothesis H15 : y2 ∈ SNoL z

L953

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L954

Hypothesis H17 : SNo x2

L955

Hypothesis H18 : y < x2

L956

Hypothesis H19 : SNo y2

L957

Hypothesis H20 : y2 < z

L958

Hypothesis H21 : SNo (g u y)

L959

Hypothesis H22 : SNo (g u x2)

L960

Hypothesis H24 : SNo (g x2 z + g y y2)

L961

Hypothesis H25 : SNo (g x2 y2)

L962

Hypothesis H26 : SNo (v + g x2 y2)

L963

Hypothesis H27 : SNo (g x (v + g x2 y2))

L964

Theorem. (Conj_mul_SNo_assoc_lem1__36__23)

SNo (g u (v + g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is a029ae... and proposition id is 2855e8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__36__27

L970

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__36__27 TMXm3oEE8WuitCqXwMMsoJar9mKvcg7PUGg bounty of about 25 bars ***)

L971

Variable x : set

L972

Variable y : set

L973

Variable z : set

L974

Variable w : set

L975

Variable u : set

L976

Variable v : set

L977

Variable x2 : set

L978

Variable y2 : set

L979

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L980

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L981

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L982

Hypothesis H3 : SNo x

L983

Hypothesis H4 : SNo y

L984

Hypothesis H5 : SNo z

L985

Hypothesis H6 : SNo (g x y)

L986

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L987

Hypothesis H8 : u ∈ SNoR x

L988

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L989

Hypothesis H10 : SNo u

L990

Hypothesis H11 : x < u

L991

Hypothesis H12 : SNo v

L992

Hypothesis H13 : SNo (g u v)

L993

Hypothesis H14 : x2 ∈ SNoR y

L994

Hypothesis H15 : y2 ∈ SNoL z

L995

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L996

Hypothesis H17 : SNo x2

L997

Hypothesis H18 : y < x2

L998

Hypothesis H19 : SNo y2

L999

Hypothesis H20 : y2 < z

L1000

Hypothesis H21 : SNo (g u y)

L1001

Hypothesis H22 : SNo (g u x2)

L1002

Hypothesis H23 : SNo (g x x2)

L1003

Hypothesis H24 : SNo (g x2 z + g y y2)

L1004

Hypothesis H25 : SNo (g x2 y2)

L1005

Hypothesis H26 : SNo (v + g x2 y2)

L1006

Theorem. (Conj_mul_SNo_assoc_lem1__36__27)

SNo (g u (v + g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 6eb7dd... and proposition id is a6d6fb...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__37__11

L1012

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__37__11 TMWAK1YiutPPtwaEHc8jUqMBob1RHvnvANu bounty of about 25 bars ***)

L1013

Variable x : set

L1014

Variable y : set

L1015

Variable z : set

L1016

Variable w : set

L1017

Variable u : set

L1018

Variable v : set

L1019

Variable x2 : set

L1020

Variable y2 : set

L1021

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1022

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1023

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1024

Hypothesis H3 : SNo x

L1025

Hypothesis H4 : SNo y

L1026

Hypothesis H5 : SNo z

L1027

Hypothesis H6 : SNo (g x y)

L1028

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1029

Hypothesis H8 : u ∈ SNoR x

L1030

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1031

Hypothesis H10 : SNo u

L1032

Hypothesis H12 : SNo v

L1033

Hypothesis H13 : SNo (g u v)

L1034

Hypothesis H14 : x2 ∈ SNoR y

L1035

Hypothesis H15 : y2 ∈ SNoL z

L1036

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1037

Hypothesis H17 : SNo x2

L1038

Hypothesis H18 : y < x2

L1039

Hypothesis H19 : SNo y2

L1040

Hypothesis H20 : y2 < z

L1041

Hypothesis H21 : SNo (g u y)

L1042

Hypothesis H22 : SNo (g u x2)

L1043

Hypothesis H23 : SNo (g x x2)

L1044

Hypothesis H24 : SNo (g x2 z + g y y2)

L1045

Hypothesis H25 : SNo (g x2 y2)

L1046

Hypothesis H26 : SNo (v + g x2 y2)

L1047

Theorem. (Conj_mul_SNo_assoc_lem1__37__11)

SNo (g x (v + g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 1aa07c... and proposition id is 0c790e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__38__11

L1053

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__38__11 TMQa9c82UPAFid5doSiewo6VTcXqJHGLjvp bounty of about 25 bars ***)

L1054

Variable x : set

L1055

Variable y : set

L1056

Variable z : set

L1057

Variable w : set

L1058

Variable u : set

L1059

Variable v : set

L1060

Variable x2 : set

L1061

Variable y2 : set

L1062

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1063

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1064

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1065

Hypothesis H3 : SNo x

L1066

Hypothesis H4 : SNo y

L1067

Hypothesis H5 : SNo z

L1068

Hypothesis H6 : SNo (g x y)

L1069

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1070

Hypothesis H8 : u ∈ SNoR x

L1071

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1072

Hypothesis H10 : SNo u

L1073

Hypothesis H12 : SNo v

L1074

Hypothesis H13 : SNo (g u v)

L1075

Hypothesis H14 : x2 ∈ SNoR y

L1076

Hypothesis H15 : y2 ∈ SNoL z

L1077

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1078

Hypothesis H17 : SNo x2

L1079

Hypothesis H18 : y < x2

L1080

Hypothesis H19 : SNo y2

L1081

Hypothesis H20 : y2 < z

L1082

Hypothesis H21 : SNo (g u y)

L1083

Hypothesis H22 : SNo (g u x2)

L1084

Hypothesis H23 : SNo (g x x2)

L1085

Hypothesis H24 : SNo (g x2 z + g y y2)

L1086

Hypothesis H25 : SNo (g x2 y2)

L1087

Theorem. (Conj_mul_SNo_assoc_lem1__38__11)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 7b7b12... and proposition id is c92dd4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__38__13

L1093

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__38__13 TMPKkBV5dLPNpNPmUwuMny9sgzLU2bev2VN bounty of about 25 bars ***)

L1094

Variable x : set

L1095

Variable y : set

L1096

Variable z : set

L1097

Variable w : set

L1098

Variable u : set

L1099

Variable v : set

L1100

Variable x2 : set

L1101

Variable y2 : set

L1102

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1103

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1104

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1105

Hypothesis H3 : SNo x

L1106

Hypothesis H4 : SNo y

L1107

Hypothesis H5 : SNo z

L1108

Hypothesis H6 : SNo (g x y)

L1109

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1110

Hypothesis H8 : u ∈ SNoR x

L1111

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1112

Hypothesis H10 : SNo u

L1113

Hypothesis H11 : x < u

L1114

Hypothesis H12 : SNo v

L1115

Hypothesis H14 : x2 ∈ SNoR y

L1116

Hypothesis H15 : y2 ∈ SNoL z

L1117

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1118

Hypothesis H17 : SNo x2

L1119

Hypothesis H18 : y < x2

L1120

Hypothesis H19 : SNo y2

L1121

Hypothesis H20 : y2 < z

L1122

Hypothesis H21 : SNo (g u y)

L1123

Hypothesis H22 : SNo (g u x2)

L1124

Hypothesis H23 : SNo (g x x2)

L1125

Hypothesis H24 : SNo (g x2 z + g y y2)

L1126

Hypothesis H25 : SNo (g x2 y2)

L1127

Theorem. (Conj_mul_SNo_assoc_lem1__38__13)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 2fc98b... and proposition id is ba3bd3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__40__19

L1133

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__40__19 TMQURrqTZ6oakmdwtmTDeD8zanuR2fgTYzK bounty of about 25 bars ***)

L1134

Variable x : set

L1135

Variable y : set

L1136

Variable z : set

L1137

Variable w : set

L1138

Variable u : set

L1139

Variable v : set

L1140

Variable x2 : set

L1141

Variable y2 : set

L1142

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1143

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1144

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1145

Hypothesis H3 : SNo x

L1146

Hypothesis H4 : SNo y

L1147

Hypothesis H5 : SNo z

L1148

Hypothesis H6 : SNo (g x y)

L1149

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1150

Hypothesis H8 : u ∈ SNoR x

L1151

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1152

Hypothesis H10 : SNo u

L1153

Hypothesis H11 : x < u

L1154

Hypothesis H12 : SNo v

L1155

Hypothesis H13 : SNo (g u v)

L1156

Hypothesis H14 : x2 ∈ SNoR y

L1157

Hypothesis H15 : y2 ∈ SNoL z

L1158

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1159

Hypothesis H17 : SNo x2

L1160

Hypothesis H18 : y < x2

L1161

Hypothesis H20 : y2 < z

L1162

Hypothesis H21 : SNo (g u y)

L1163

Hypothesis H22 : SNo (g u x2)

L1164

Hypothesis H23 : SNo (g x x2)

L1165

Hypothesis H24 : SNo (g x2 z)

L1166

Hypothesis H25 : SNo (g y y2)

L1167

Theorem. (Conj_mul_SNo_assoc_lem1__40__19)

SNo (g x2 z + g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is be48c1... and proposition id is ee6363...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__40__25

L1173

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__40__25 TMTJjwjowoSyPq2g11anjQqxBe9UKx44qKh bounty of about 25 bars ***)

L1174

Variable x : set

L1175

Variable y : set

L1176

Variable z : set

L1177

Variable w : set

L1178

Variable u : set

L1179

Variable v : set

L1180

Variable x2 : set

L1181

Variable y2 : set

L1182

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1183

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1184

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1185

Hypothesis H3 : SNo x

L1186

Hypothesis H4 : SNo y

L1187

Hypothesis H5 : SNo z

L1188

Hypothesis H6 : SNo (g x y)

L1189

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1190

Hypothesis H8 : u ∈ SNoR x

L1191

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1192

Hypothesis H10 : SNo u

L1193

Hypothesis H11 : x < u

L1194

Hypothesis H12 : SNo v

L1195

Hypothesis H13 : SNo (g u v)

L1196

Hypothesis H14 : x2 ∈ SNoR y

L1197

Hypothesis H15 : y2 ∈ SNoL z

L1198

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1199

Hypothesis H17 : SNo x2

L1200

Hypothesis H18 : y < x2

L1201

Hypothesis H19 : SNo y2

L1202

Hypothesis H20 : y2 < z

L1203

Hypothesis H21 : SNo (g u y)

L1204

Hypothesis H22 : SNo (g u x2)

L1205

Hypothesis H23 : SNo (g x x2)

L1206

Hypothesis H24 : SNo (g x2 z)

L1207

Theorem. (Conj_mul_SNo_assoc_lem1__40__25)

SNo (g x2 z + g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is d17bb2... and proposition id is c394a2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__41__17

L1213

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__41__17 TMd9LpRX8gAnDacQVm9cH4mazv6L8ZYgPWd bounty of about 25 bars ***)

L1214

Variable x : set

L1215

Variable y : set

L1216

Variable z : set

L1217

Variable w : set

L1218

Variable u : set

L1219

Variable v : set

L1220

Variable x2 : set

L1221

Variable y2 : set

L1222

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1223

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1224

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1225

Hypothesis H3 : SNo x

L1226

Hypothesis H4 : SNo y

L1227

Hypothesis H5 : SNo z

L1228

Hypothesis H6 : SNo (g x y)

L1229

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1230

Hypothesis H8 : u ∈ SNoR x

L1231

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1232

Hypothesis H10 : SNo u

L1233

Hypothesis H11 : x < u

L1234

Hypothesis H12 : SNo v

L1235

Hypothesis H13 : SNo (g u v)

L1236

Hypothesis H14 : x2 ∈ SNoR y

L1237

Hypothesis H15 : y2 ∈ SNoL z

L1238

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1239

Hypothesis H18 : y < x2

L1240

Hypothesis H19 : SNo y2

L1241

Hypothesis H20 : y2 < z

L1242

Hypothesis H21 : SNo (g u y)

L1243

Hypothesis H22 : SNo (g u x2)

L1244

Hypothesis H23 : SNo (g x x2)

L1245

Hypothesis H24 : SNo (g x2 z)

L1246

Theorem. (Conj_mul_SNo_assoc_lem1__41__17)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is cea3ef... and proposition id is 130018...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__43__3

L1252

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__43__3 TMLuxqALyfEYoy11UpMfpZKejHLCds3ZHzB bounty of about 25 bars ***)

L1253

Variable x : set

L1254

Variable y : set

L1255

Variable z : set

L1256

Variable w : set

L1257

Variable u : set

L1258

Variable v : set

L1259

Variable x2 : set

L1260

Variable y2 : set

L1261

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1262

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1263

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1264

Hypothesis H4 : SNo y

L1265

Hypothesis H5 : SNo z

L1266

Hypothesis H6 : SNo (g x y)

L1267

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1268

Hypothesis H8 : u ∈ SNoR x

L1269

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1270

Hypothesis H10 : SNo u

L1271

Hypothesis H11 : x < u

L1272

Hypothesis H12 : SNo v

L1273

Hypothesis H13 : SNo (g u v)

L1274

Hypothesis H14 : x2 ∈ SNoR y

L1275

Hypothesis H15 : y2 ∈ SNoL z

L1276

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1277

Hypothesis H17 : SNo x2

L1278

Hypothesis H18 : y < x2

L1279

Hypothesis H19 : SNo y2

L1280

Hypothesis H20 : y2 < z

L1281

Hypothesis H21 : SNo (g u y)

L1282

Hypothesis H22 : SNo (g u x2)

L1283

Theorem. (Conj_mul_SNo_assoc_lem1__43__3)

SNo (g x x2) → w < g (g x y) z

In Proofgold the corresponding term root is c15325... and proposition id is 361f69...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__43__6

L1289

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__43__6 TMK913J4oFuZMfBVFUpEnsC3idSWGxyauUx bounty of about 25 bars ***)

L1290

Variable x : set

L1291

Variable y : set

L1292

Variable z : set

L1293

Variable w : set

L1294

Variable u : set

L1295

Variable v : set

L1296

Variable x2 : set

L1297

Variable y2 : set

L1298

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1299

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1300

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1301

Hypothesis H3 : SNo x

L1302

Hypothesis H4 : SNo y

L1303

Hypothesis H5 : SNo z

L1304

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1305

Hypothesis H8 : u ∈ SNoR x

L1306

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1307

Hypothesis H10 : SNo u

L1308

Hypothesis H11 : x < u

L1309

Hypothesis H12 : SNo v

L1310

Hypothesis H13 : SNo (g u v)

L1311

Hypothesis H14 : x2 ∈ SNoR y

L1312

Hypothesis H15 : y2 ∈ SNoL z

L1313

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1314

Hypothesis H17 : SNo x2

L1315

Hypothesis H18 : y < x2

L1316

Hypothesis H19 : SNo y2

L1317

Hypothesis H20 : y2 < z

L1318

Hypothesis H21 : SNo (g u y)

L1319

Hypothesis H22 : SNo (g u x2)

L1320

Theorem. (Conj_mul_SNo_assoc_lem1__43__6)

SNo (g x x2) → w < g (g x y) z

In Proofgold the corresponding term root is e0a2d1... and proposition id is fa1c3b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__43__19

L1326

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__43__19 TMR12FQzUEso6WDMABMSebV8XBA2NUA7ubd bounty of about 25 bars ***)

L1327

Variable x : set

L1328

Variable y : set

L1329

Variable z : set

L1330

Variable w : set

L1331

Variable u : set

L1332

Variable v : set

L1333

Variable x2 : set

L1334

Variable y2 : set

L1335

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1336

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1337

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1338

Hypothesis H3 : SNo x

L1339

Hypothesis H4 : SNo y

L1340

Hypothesis H5 : SNo z

L1341

Hypothesis H6 : SNo (g x y)

L1342

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1343

Hypothesis H8 : u ∈ SNoR x

L1344

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1345

Hypothesis H10 : SNo u

L1346

Hypothesis H11 : x < u

L1347

Hypothesis H12 : SNo v

L1348

Hypothesis H13 : SNo (g u v)

L1349

Hypothesis H14 : x2 ∈ SNoR y

L1350

Hypothesis H15 : y2 ∈ SNoL z

L1351

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1352

Hypothesis H17 : SNo x2

L1353

Hypothesis H18 : y < x2

L1354

Hypothesis H20 : y2 < z

L1355

Hypothesis H21 : SNo (g u y)

L1356

Hypothesis H22 : SNo (g u x2)

L1357

Theorem. (Conj_mul_SNo_assoc_lem1__43__19)

SNo (g x x2) → w < g (g x y) z

In Proofgold the corresponding term root is 3fa025... and proposition id is b44bb1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__46__0

L1363

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__46__0 TMLrdp434475c5f4HSZnrVnbVMuop74Hemb bounty of about 25 bars ***)

L1364

Variable x : set

L1365

Variable y : set

L1366

Variable z : set

L1367

Variable w : set

L1368

Variable u : set

L1369

Variable v : set

L1370

Variable x2 : set

L1371

Variable y2 : set

L1372

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1373

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1374

Hypothesis H3 : SNo x

L1375

Hypothesis H4 : SNo y

L1376

Hypothesis H5 : SNo z

L1377

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1378

Hypothesis H7 : u ∈ SNoR x

L1379

Hypothesis H8 : w = g u (g y z) + g x v + - (g u v)

L1380

Hypothesis H9 : SNo u

L1381

Hypothesis H10 : x < u

L1382

Hypothesis H11 : SNo v

L1383

Hypothesis H12 : x2 ∈ SNoL y

L1384

Hypothesis H13 : y2 ∈ SNoR z

L1385

Hypothesis H14 : (g x2 z + g y y2) ≤ v + g x2 y2

L1386

Hypothesis H15 : SNo x2

L1387

Hypothesis H16 : x2 < y

L1388

Hypothesis H17 : SNo y2

L1389

Hypothesis H18 : z < y2

L1390

Hypothesis H19 : SNo (g x2 z + g y y2)

L1391

Hypothesis H20 : SNo (v + g x2 y2)

L1392

Hypothesis H21 : SNo (g x (v + g x2 y2))

L1393

Hypothesis H22 : SNo (g u (v + g x2 y2))

L1394

Hypothesis H23 : SNo (g u (g x2 z + g y y2))

L1395

Hypothesis H24 : SNo (g x (g x2 z + g y y2))

L1396

Hypothesis H25 : SNo (g u y + g x x2)

L1397

Hypothesis H26 : SNo (g x y + g u x2)

L1398

Hypothesis H27 : SNo (g (g u y + g x x2) z)

L1399

Hypothesis H28 : SNo (g (g x y + g u x2) z)

L1400

Hypothesis H29 : SNo (g (g u y + g x x2) y2)

L1401

Theorem. (Conj_mul_SNo_assoc_lem1__46__0)

SNo (g (g x y + g u x2) y2) → w < g (g x y) z

In Proofgold the corresponding term root is 1cc964... and proposition id is f39a11...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__47__5

L1407

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__47__5 TMaXoKJCyF4UPnYAAPK5CKv71aAvjxMQjE9 bounty of about 25 bars ***)

L1408

Variable x : set

L1409

Variable y : set

L1410

Variable z : set

L1411

Variable w : set

L1412

Variable u : set

L1413

Variable v : set

L1414

Variable x2 : set

L1415

Variable y2 : set

L1416

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1417

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1418

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1419

Hypothesis H3 : SNo x

L1420

Hypothesis H4 : SNo y

L1421

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1422

Hypothesis H7 : u ∈ SNoR x

L1423

Hypothesis H8 : w = g u (g y z) + g x v + - (g u v)

L1424

Hypothesis H9 : SNo u

L1425

Hypothesis H10 : x < u

L1426

Hypothesis H11 : SNo v

L1427

Hypothesis H12 : x2 ∈ SNoL y

L1428

Hypothesis H13 : y2 ∈ SNoR z

L1429

Hypothesis H14 : (g x2 z + g y y2) ≤ v + g x2 y2

L1430

Hypothesis H15 : SNo x2

L1431

Hypothesis H16 : x2 < y

L1432

Hypothesis H17 : SNo y2

L1433

Hypothesis H18 : z < y2

L1434

Hypothesis H19 : SNo (g x2 z + g y y2)

L1435

Hypothesis H20 : SNo (v + g x2 y2)

L1436

Hypothesis H21 : SNo (g x (v + g x2 y2))

L1437

Hypothesis H22 : SNo (g u (v + g x2 y2))

L1438

Hypothesis H23 : SNo (g u (g x2 z + g y y2))

L1439

Hypothesis H24 : SNo (g x (g x2 z + g y y2))

L1440

Hypothesis H25 : SNo (g u y + g x x2)

L1441

Hypothesis H26 : SNo (g x y + g u x2)

L1442

Hypothesis H27 : SNo (g (g u y + g x x2) z)

L1443

Hypothesis H28 : SNo (g (g x y + g u x2) z)

L1444

Theorem. (Conj_mul_SNo_assoc_lem1__47__5)

SNo (g (g u y + g x x2) y2) → w < g (g x y) z

In Proofgold the corresponding term root is 934560... and proposition id is 4a86ce...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__48__1

L1450

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__48__1 TMFV4jNtq97tZkzqQ6wSz3TMyjiRKSv6u4G bounty of about 25 bars ***)

L1451

Variable x : set

L1452

Variable y : set

L1453

Variable z : set

L1454

Variable w : set

L1455

Variable u : set

L1456

Variable v : set

L1457

Variable x2 : set

L1458

Variable y2 : set

L1459

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1460

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1461

Hypothesis H3 : SNo x

L1462

Hypothesis H4 : SNo y

L1463

Hypothesis H5 : SNo z

L1464

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1465

Hypothesis H7 : u ∈ SNoR x

L1466

Hypothesis H8 : w = g u (g y z) + g x v + - (g u v)

L1467

Hypothesis H9 : SNo u

L1468

Hypothesis H10 : x < u

L1469

Hypothesis H11 : SNo v

L1470

Hypothesis H12 : x2 ∈ SNoL y

L1471

Hypothesis H13 : y2 ∈ SNoR z

L1472

Hypothesis H14 : (g x2 z + g y y2) ≤ v + g x2 y2

L1473

Hypothesis H15 : SNo x2

L1474

Hypothesis H16 : x2 < y

L1475

Hypothesis H17 : SNo y2

L1476

Hypothesis H18 : z < y2

L1477

Hypothesis H19 : SNo (g x2 z + g y y2)

L1478

Hypothesis H20 : SNo (v + g x2 y2)

L1479

Hypothesis H21 : SNo (g x (v + g x2 y2))

L1480

Hypothesis H22 : SNo (g u (v + g x2 y2))

L1481

Hypothesis H23 : SNo (g u (g x2 z + g y y2))

L1482

Hypothesis H24 : SNo (g x (g x2 z + g y y2))

L1483

Hypothesis H25 : SNo (g u y + g x x2)

L1484

Hypothesis H26 : SNo (g x y + g u x2)

L1485

Hypothesis H27 : SNo (g (g u y + g x x2) z)

L1486

Theorem. (Conj_mul_SNo_assoc_lem1__48__1)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is e6b7c0... and proposition id is c331ef...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__48__22

L1492

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__48__22 TMNSWLXRZ6wC9d5ogaSiuFsapX7M3qNDu5H bounty of about 25 bars ***)

L1493

Variable x : set

L1494

Variable y : set

L1495

Variable z : set

L1496

Variable w : set

L1497

Variable u : set

L1498

Variable v : set

L1499

Variable x2 : set

L1500

Variable y2 : set

L1501

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1502

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1503

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1504

Hypothesis H3 : SNo x

L1505

Hypothesis H4 : SNo y

L1506

Hypothesis H5 : SNo z

L1507

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1508

Hypothesis H7 : u ∈ SNoR x

L1509

Hypothesis H8 : w = g u (g y z) + g x v + - (g u v)

L1510

Hypothesis H9 : SNo u

L1511

Hypothesis H10 : x < u

L1512

Hypothesis H11 : SNo v

L1513

Hypothesis H12 : x2 ∈ SNoL y

L1514

Hypothesis H13 : y2 ∈ SNoR z

L1515

Hypothesis H14 : (g x2 z + g y y2) ≤ v + g x2 y2

L1516

Hypothesis H15 : SNo x2

L1517

Hypothesis H16 : x2 < y

L1518

Hypothesis H17 : SNo y2

L1519

Hypothesis H18 : z < y2

L1520

Hypothesis H19 : SNo (g x2 z + g y y2)

L1521

Hypothesis H20 : SNo (v + g x2 y2)

L1522

Hypothesis H21 : SNo (g x (v + g x2 y2))

L1523

Hypothesis H23 : SNo (g u (g x2 z + g y y2))

L1524

Hypothesis H24 : SNo (g x (g x2 z + g y y2))

L1525

Hypothesis H25 : SNo (g u y + g x x2)

L1526

Hypothesis H26 : SNo (g x y + g u x2)

L1527

Hypothesis H27 : SNo (g (g u y + g x x2) z)

L1528

Theorem. (Conj_mul_SNo_assoc_lem1__48__22)

SNo (g (g x y + g u x2) z) → w < g (g x y) z

In Proofgold the corresponding term root is 9bf01f... and proposition id is 2eb556...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__50__11

L1534

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__50__11 TMNNjV6TSwfi7Uhy2et3mbD98DHCiXvFUDc bounty of about 25 bars ***)

L1535

Variable x : set

L1536

Variable y : set

L1537

Variable z : set

L1538

Variable w : set

L1539

Variable u : set

L1540

Variable v : set

L1541

Variable x2 : set

L1542

Variable y2 : set

L1543

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1544

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1545

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1546

Hypothesis H3 : SNo x

L1547

Hypothesis H4 : SNo y

L1548

Hypothesis H5 : SNo z

L1549

Hypothesis H6 : SNo (g x y)

L1550

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1551

Hypothesis H8 : u ∈ SNoR x

L1552

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1553

Hypothesis H10 : SNo u

L1554

Hypothesis H12 : SNo v

L1555

Hypothesis H13 : x2 ∈ SNoL y

L1556

Hypothesis H14 : y2 ∈ SNoR z

L1557

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L1558

Hypothesis H16 : SNo x2

L1559

Hypothesis H17 : x2 < y

L1560

Hypothesis H18 : SNo y2

L1561

Hypothesis H19 : z < y2

L1562

Hypothesis H20 : SNo (g u x2)

L1563

Hypothesis H21 : SNo (g x2 z + g y y2)

L1564

Hypothesis H22 : SNo (v + g x2 y2)

L1565

Hypothesis H23 : SNo (g x (v + g x2 y2))

L1566

Hypothesis H24 : SNo (g u (v + g x2 y2))

L1567

Hypothesis H25 : SNo (g u (g x2 z + g y y2))

L1568

Hypothesis H26 : SNo (g x (g x2 z + g y y2))

L1569

Hypothesis H27 : SNo (g u y + g x x2)

L1570

Theorem. (Conj_mul_SNo_assoc_lem1__50__11)

SNo (g x y + g u x2) → w < g (g x y) z

In Proofgold the corresponding term root is 1141e6... and proposition id is 2e775e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__52__3

L1576

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__52__3 TMTSHJyN58HYTk3dU1hKDXB56j3A3r4u7ww bounty of about 25 bars ***)

L1577

Variable x : set

L1578

Variable y : set

L1579

Variable z : set

L1580

Variable w : set

L1581

Variable u : set

L1582

Variable v : set

L1583

Variable x2 : set

L1584

Variable y2 : set

L1585

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1586

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1587

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1588

Hypothesis H4 : SNo y

L1589

Hypothesis H5 : SNo z

L1590

Hypothesis H6 : SNo (g x y)

L1591

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1592

Hypothesis H8 : u ∈ SNoR x

L1593

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1594

Hypothesis H10 : SNo u

L1595

Hypothesis H11 : x < u

L1596

Hypothesis H12 : SNo v

L1597

Hypothesis H13 : x2 ∈ SNoL y

L1598

Hypothesis H14 : y2 ∈ SNoR z

L1599

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L1600

Hypothesis H16 : SNo x2

L1601

Hypothesis H17 : x2 < y

L1602

Hypothesis H18 : SNo y2

L1603

Hypothesis H19 : z < y2

L1604

Hypothesis H20 : SNo (g u y)

L1605

Hypothesis H21 : SNo (g u x2)

L1606

Hypothesis H22 : SNo (g x x2)

L1607

Hypothesis H23 : SNo (g x2 z + g y y2)

L1608

Hypothesis H24 : SNo (v + g x2 y2)

L1609

Hypothesis H25 : SNo (g x (v + g x2 y2))

L1610

Hypothesis H26 : SNo (g u (v + g x2 y2))

L1611

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L1612

Theorem. (Conj_mul_SNo_assoc_lem1__52__3)

SNo (g x (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 8891dc... and proposition id is 976c93...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__52__10

L1618

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__52__10 TMUkHcaUW7PQRPSy1hihi9jYuSnBvSjBERq bounty of about 25 bars ***)

L1619

Variable x : set

L1620

Variable y : set

L1621

Variable z : set

L1622

Variable w : set

L1623

Variable u : set

L1624

Variable v : set

L1625

Variable x2 : set

L1626

Variable y2 : set

L1627

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1628

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1629

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1630

Hypothesis H3 : SNo x

L1631

Hypothesis H4 : SNo y

L1632

Hypothesis H5 : SNo z

L1633

Hypothesis H6 : SNo (g x y)

L1634

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1635

Hypothesis H8 : u ∈ SNoR x

L1636

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1637

Hypothesis H11 : x < u

L1638

Hypothesis H12 : SNo v

L1639

Hypothesis H13 : x2 ∈ SNoL y

L1640

Hypothesis H14 : y2 ∈ SNoR z

L1641

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L1642

Hypothesis H16 : SNo x2

L1643

Hypothesis H17 : x2 < y

L1644

Hypothesis H18 : SNo y2

L1645

Hypothesis H19 : z < y2

L1646

Hypothesis H20 : SNo (g u y)

L1647

Hypothesis H21 : SNo (g u x2)

L1648

Hypothesis H22 : SNo (g x x2)

L1649

Hypothesis H23 : SNo (g x2 z + g y y2)

L1650

Hypothesis H24 : SNo (v + g x2 y2)

L1651

Hypothesis H25 : SNo (g x (v + g x2 y2))

L1652

Hypothesis H26 : SNo (g u (v + g x2 y2))

L1653

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L1654

Theorem. (Conj_mul_SNo_assoc_lem1__52__10)

SNo (g x (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 358d81... and proposition id is 9c7fb3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__52__25

L1660

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__52__25 TMd11PPv8pwxY588eVpd2qqHhqbzkBQWpLP bounty of about 25 bars ***)

L1661

Variable x : set

L1662

Variable y : set

L1663

Variable z : set

L1664

Variable w : set

L1665

Variable u : set

L1666

Variable v : set

L1667

Variable x2 : set

L1668

Variable y2 : set

L1669

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1670

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1671

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1672

Hypothesis H3 : SNo x

L1673

Hypothesis H4 : SNo y

L1674

Hypothesis H5 : SNo z

L1675

Hypothesis H6 : SNo (g x y)

L1676

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1677

Hypothesis H8 : u ∈ SNoR x

L1678

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1679

Hypothesis H10 : SNo u

L1680

Hypothesis H11 : x < u

L1681

Hypothesis H12 : SNo v

L1682

Hypothesis H13 : x2 ∈ SNoL y

L1683

Hypothesis H14 : y2 ∈ SNoR z

L1684

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L1685

Hypothesis H16 : SNo x2

L1686

Hypothesis H17 : x2 < y

L1687

Hypothesis H18 : SNo y2

L1688

Hypothesis H19 : z < y2

L1689

Hypothesis H20 : SNo (g u y)

L1690

Hypothesis H21 : SNo (g u x2)

L1691

Hypothesis H22 : SNo (g x x2)

L1692

Hypothesis H23 : SNo (g x2 z + g y y2)

L1693

Hypothesis H24 : SNo (v + g x2 y2)

L1694

Hypothesis H26 : SNo (g u (v + g x2 y2))

L1695

Hypothesis H27 : SNo (g u (g x2 z + g y y2))

L1696

Theorem. (Conj_mul_SNo_assoc_lem1__52__25)

SNo (g x (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is c9ac54... and proposition id is 6cf6f3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__53__6

L1702

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__53__6 TMMCn2CEmYTwJMx2Jqx1QGJ2wMDm1fZpb9w bounty of about 25 bars ***)

L1703

Variable x : set

L1704

Variable y : set

L1705

Variable z : set

L1706

Variable w : set

L1707

Variable u : set

L1708

Variable v : set

L1709

Variable x2 : set

L1710

Variable y2 : set

L1711

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1712

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1713

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1714

Hypothesis H3 : SNo x

L1715

Hypothesis H4 : SNo y

L1716

Hypothesis H5 : SNo z

L1717

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1718

Hypothesis H8 : u ∈ SNoR x

L1719

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1720

Hypothesis H10 : SNo u

L1721

Hypothesis H11 : x < u

L1722

Hypothesis H12 : SNo v

L1723

Hypothesis H13 : SNo (g u v)

L1724

Hypothesis H14 : x2 ∈ SNoL y

L1725

Hypothesis H15 : y2 ∈ SNoR z

L1726

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1727

Hypothesis H17 : SNo x2

L1728

Hypothesis H18 : x2 < y

L1729

Hypothesis H19 : SNo y2

L1730

Hypothesis H20 : z < y2

L1731

Hypothesis H21 : SNo (g u y)

L1732

Hypothesis H22 : SNo (g u x2)

L1733

Hypothesis H23 : SNo (g x x2)

L1734

Hypothesis H24 : SNo (g x2 z + g y y2)

L1735

Hypothesis H25 : SNo (v + g x2 y2)

L1736

Hypothesis H26 : SNo (g x (v + g x2 y2))

L1737

Hypothesis H27 : SNo (g u (v + g x2 y2))

L1738

Hypothesis H28 : SNo (g u (g x2 y2))

L1739

Hypothesis H29 : SNo (g u (g x2 z + g y y2))

L1740

Theorem. (Conj_mul_SNo_assoc_lem1__53__6)

SNo (g u v + g u (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 587546... and proposition id is 9ae5ea...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__53__23

L1746

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__53__23 TMcf7WjNgDsKWb4xNurDQukYakpBiFNJZ4Q bounty of about 25 bars ***)

L1747

Variable x : set

L1748

Variable y : set

L1749

Variable z : set

L1750

Variable w : set

L1751

Variable u : set

L1752

Variable v : set

L1753

Variable x2 : set

L1754

Variable y2 : set

L1755

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1756

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1757

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1758

Hypothesis H3 : SNo x

L1759

Hypothesis H4 : SNo y

L1760

Hypothesis H5 : SNo z

L1761

Hypothesis H6 : SNo (g x y)

L1762

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1763

Hypothesis H8 : u ∈ SNoR x

L1764

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1765

Hypothesis H10 : SNo u

L1766

Hypothesis H11 : x < u

L1767

Hypothesis H12 : SNo v

L1768

Hypothesis H13 : SNo (g u v)

L1769

Hypothesis H14 : x2 ∈ SNoL y

L1770

Hypothesis H15 : y2 ∈ SNoR z

L1771

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1772

Hypothesis H17 : SNo x2

L1773

Hypothesis H18 : x2 < y

L1774

Hypothesis H19 : SNo y2

L1775

Hypothesis H20 : z < y2

L1776

Hypothesis H21 : SNo (g u y)

L1777

Hypothesis H22 : SNo (g u x2)

L1778

Hypothesis H24 : SNo (g x2 z + g y y2)

L1779

Hypothesis H25 : SNo (v + g x2 y2)

L1780

Hypothesis H26 : SNo (g x (v + g x2 y2))

L1781

Hypothesis H27 : SNo (g u (v + g x2 y2))

L1782

Hypothesis H28 : SNo (g u (g x2 y2))

L1783

Hypothesis H29 : SNo (g u (g x2 z + g y y2))

L1784

Theorem. (Conj_mul_SNo_assoc_lem1__53__23)

SNo (g u v + g u (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is a2730d... and proposition id is 3d0157...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__54__13

L1790

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__54__13 TMNZsnirKwo5MDU3KNRDDdjWqNMB7ViT3UY bounty of about 25 bars ***)

L1791

Variable x : set

L1792

Variable y : set

L1793

Variable z : set

L1794

Variable w : set

L1795

Variable u : set

L1796

Variable v : set

L1797

Variable x2 : set

L1798

Variable y2 : set

L1799

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1800

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1801

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1802

Hypothesis H3 : SNo x

L1803

Hypothesis H4 : SNo y

L1804

Hypothesis H5 : SNo z

L1805

Hypothesis H6 : SNo (g x y)

L1806

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1807

Hypothesis H8 : u ∈ SNoR x

L1808

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1809

Hypothesis H10 : SNo u

L1810

Hypothesis H11 : x < u

L1811

Hypothesis H12 : SNo v

L1812

Hypothesis H14 : x2 ∈ SNoL y

L1813

Hypothesis H15 : y2 ∈ SNoR z

L1814

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1815

Hypothesis H17 : SNo x2

L1816

Hypothesis H18 : x2 < y

L1817

Hypothesis H19 : SNo y2

L1818

Hypothesis H20 : z < y2

L1819

Hypothesis H21 : SNo (g u y)

L1820

Hypothesis H22 : SNo (g u x2)

L1821

Hypothesis H23 : SNo (g x x2)

L1822

Hypothesis H24 : SNo (g x2 z + g y y2)

L1823

Hypothesis H25 : SNo (v + g x2 y2)

L1824

Hypothesis H26 : SNo (g x (v + g x2 y2))

L1825

Hypothesis H27 : SNo (g u (v + g x2 y2))

L1826

Hypothesis H28 : SNo (g u (g x2 y2))

L1827

Theorem. (Conj_mul_SNo_assoc_lem1__54__13)

SNo (g u (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 61284d... and proposition id is 885e11...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__54__20

L1833

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__54__20 TMFm628uzXFmwooviQJocSN9HbPcXXwTycV bounty of about 25 bars ***)

L1834

Variable x : set

L1835

Variable y : set

L1836

Variable z : set

L1837

Variable w : set

L1838

Variable u : set

L1839

Variable v : set

L1840

Variable x2 : set

L1841

Variable y2 : set

L1842

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1843

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1844

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1845

Hypothesis H3 : SNo x

L1846

Hypothesis H4 : SNo y

L1847

Hypothesis H5 : SNo z

L1848

Hypothesis H6 : SNo (g x y)

L1849

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1850

Hypothesis H8 : u ∈ SNoR x

L1851

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1852

Hypothesis H10 : SNo u

L1853

Hypothesis H11 : x < u

L1854

Hypothesis H12 : SNo v

L1855

Hypothesis H13 : SNo (g u v)

L1856

Hypothesis H14 : x2 ∈ SNoL y

L1857

Hypothesis H15 : y2 ∈ SNoR z

L1858

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1859

Hypothesis H17 : SNo x2

L1860

Hypothesis H18 : x2 < y

L1861

Hypothesis H19 : SNo y2

L1862

Hypothesis H21 : SNo (g u y)

L1863

Hypothesis H22 : SNo (g u x2)

L1864

Hypothesis H23 : SNo (g x x2)

L1865

Hypothesis H24 : SNo (g x2 z + g y y2)

L1866

Hypothesis H25 : SNo (v + g x2 y2)

L1867

Hypothesis H26 : SNo (g x (v + g x2 y2))

L1868

Hypothesis H27 : SNo (g u (v + g x2 y2))

L1869

Hypothesis H28 : SNo (g u (g x2 y2))

L1870

Theorem. (Conj_mul_SNo_assoc_lem1__54__20)

SNo (g u (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 7274ec... and proposition id is aed699...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__54__23

L1876

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__54__23 TMFAoeDS77TbJETtBx6uZnXYbtkG5AQq9co bounty of about 25 bars ***)

L1877

Variable x : set

L1878

Variable y : set

L1879

Variable z : set

L1880

Variable w : set

L1881

Variable u : set

L1882

Variable v : set

L1883

Variable x2 : set

L1884

Variable y2 : set

L1885

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1886

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1887

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1888

Hypothesis H3 : SNo x

L1889

Hypothesis H4 : SNo y

L1890

Hypothesis H5 : SNo z

L1891

Hypothesis H6 : SNo (g x y)

L1892

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1893

Hypothesis H8 : u ∈ SNoR x

L1894

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1895

Hypothesis H10 : SNo u

L1896

Hypothesis H11 : x < u

L1897

Hypothesis H12 : SNo v

L1898

Hypothesis H13 : SNo (g u v)

L1899

Hypothesis H14 : x2 ∈ SNoL y

L1900

Hypothesis H15 : y2 ∈ SNoR z

L1901

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1902

Hypothesis H17 : SNo x2

L1903

Hypothesis H18 : x2 < y

L1904

Hypothesis H19 : SNo y2

L1905

Hypothesis H20 : z < y2

L1906

Hypothesis H21 : SNo (g u y)

L1907

Hypothesis H22 : SNo (g u x2)

L1908

Hypothesis H24 : SNo (g x2 z + g y y2)

L1909

Hypothesis H25 : SNo (v + g x2 y2)

L1910

Hypothesis H26 : SNo (g x (v + g x2 y2))

L1911

Hypothesis H27 : SNo (g u (v + g x2 y2))

L1912

Hypothesis H28 : SNo (g u (g x2 y2))

L1913

Theorem. (Conj_mul_SNo_assoc_lem1__54__23)

SNo (g u (g x2 z + g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 00273c... and proposition id is 2770ad...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__55__8

L1919

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__55__8 TMY97FH4aEH3MJdcwKPCe3EUjRi6yWb3ARr bounty of about 25 bars ***)

L1920

Variable x : set

L1921

Variable y : set

L1922

Variable z : set

L1923

Variable w : set

L1924

Variable u : set

L1925

Variable v : set

L1926

Variable x2 : set

L1927

Variable y2 : set

L1928

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1929

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1930

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1931

Hypothesis H3 : SNo x

L1932

Hypothesis H4 : SNo y

L1933

Hypothesis H5 : SNo z

L1934

Hypothesis H6 : SNo (g x y)

L1935

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1936

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1937

Hypothesis H10 : SNo u

L1938

Hypothesis H11 : x < u

L1939

Hypothesis H12 : SNo v

L1940

Hypothesis H13 : SNo (g u v)

L1941

Hypothesis H14 : x2 ∈ SNoL y

L1942

Hypothesis H15 : y2 ∈ SNoR z

L1943

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1944

Hypothesis H17 : SNo x2

L1945

Hypothesis H18 : x2 < y

L1946

Hypothesis H19 : SNo y2

L1947

Hypothesis H20 : z < y2

L1948

Hypothesis H21 : SNo (g u y)

L1949

Hypothesis H22 : SNo (g u x2)

L1950

Hypothesis H23 : SNo (g x x2)

L1951

Hypothesis H24 : SNo (g x2 z + g y y2)

L1952

Hypothesis H25 : SNo (g x2 y2)

L1953

Hypothesis H26 : SNo (v + g x2 y2)

L1954

Hypothesis H27 : SNo (g x (v + g x2 y2))

L1955

Hypothesis H28 : SNo (g u (v + g x2 y2))

L1956

Theorem. (Conj_mul_SNo_assoc_lem1__55__8)

SNo (g u (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 77236f... and proposition id is 9d009c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__57__15

L1962

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__57__15 TMYooweT4sD7pdHQ9WknUBGhp2Hi7G7xCna bounty of about 25 bars ***)

L1963

Variable x : set

L1964

Variable y : set

L1965

Variable z : set

L1966

Variable w : set

L1967

Variable u : set

L1968

Variable v : set

L1969

Variable x2 : set

L1970

Variable y2 : set

L1971

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L1972

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L1973

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L1974

Hypothesis H3 : SNo x

L1975

Hypothesis H4 : SNo y

L1976

Hypothesis H5 : SNo z

L1977

Hypothesis H6 : SNo (g x y)

L1978

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L1979

Hypothesis H8 : u ∈ SNoR x

L1980

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L1981

Hypothesis H10 : SNo u

L1982

Hypothesis H11 : x < u

L1983

Hypothesis H12 : SNo v

L1984

Hypothesis H13 : SNo (g u v)

L1985

Hypothesis H14 : x2 ∈ SNoL y

L1986

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L1987

Hypothesis H17 : SNo x2

L1988

Hypothesis H18 : x2 < y

L1989

Hypothesis H19 : SNo y2

L1990

Hypothesis H20 : z < y2

L1991

Hypothesis H21 : SNo (g u y)

L1992

Hypothesis H22 : SNo (g u x2)

L1993

Hypothesis H23 : SNo (g x x2)

L1994

Hypothesis H24 : SNo (g x2 z + g y y2)

L1995

Hypothesis H25 : SNo (g x2 y2)

L1996

Hypothesis H26 : SNo (v + g x2 y2)

L1997

Theorem. (Conj_mul_SNo_assoc_lem1__57__15)

SNo (g x (v + g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 068121... and proposition id is c4280e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__58__7

L2003

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__58__7 TMMs5P2LPAVPbzUJ5PM1QBLvb4rtViLVcqf bounty of about 25 bars ***)

L2004

Variable x : set

L2005

Variable y : set

L2006

Variable z : set

L2007

Variable w : set

L2008

Variable u : set

L2009

Variable v : set

L2010

Variable x2 : set

L2011

Variable y2 : set

L2012

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2013

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2014

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2015

Hypothesis H3 : SNo x

L2016

Hypothesis H4 : SNo y

L2017

Hypothesis H5 : SNo z

L2018

Hypothesis H6 : SNo (g x y)

L2019

Hypothesis H8 : u ∈ SNoR x

L2020

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2021

Hypothesis H10 : SNo u

L2022

Hypothesis H11 : x < u

L2023

Hypothesis H12 : SNo v

L2024

Hypothesis H13 : SNo (g u v)

L2025

Hypothesis H14 : x2 ∈ SNoL y

L2026

Hypothesis H15 : y2 ∈ SNoR z

L2027

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2028

Hypothesis H17 : SNo x2

L2029

Hypothesis H18 : x2 < y

L2030

Hypothesis H19 : SNo y2

L2031

Hypothesis H20 : z < y2

L2032

Hypothesis H21 : SNo (g u y)

L2033

Hypothesis H22 : SNo (g u x2)

L2034

Hypothesis H23 : SNo (g x x2)

L2035

Hypothesis H24 : SNo (g x2 z + g y y2)

L2036

Hypothesis H25 : SNo (g x2 y2)

L2037

Theorem. (Conj_mul_SNo_assoc_lem1__58__7)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 4f2880... and proposition id is 44e496...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__58__9

L2043

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__58__9 TMQCreWjs9eNHcMUpDz1FMJTNoy46Uvh8pn bounty of about 25 bars ***)

L2044

Variable x : set

L2045

Variable y : set

L2046

Variable z : set

L2047

Variable w : set

L2048

Variable u : set

L2049

Variable v : set

L2050

Variable x2 : set

L2051

Variable y2 : set

L2052

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2053

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2054

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2055

Hypothesis H3 : SNo x

L2056

Hypothesis H4 : SNo y

L2057

Hypothesis H5 : SNo z

L2058

Hypothesis H6 : SNo (g x y)

L2059

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2060

Hypothesis H8 : u ∈ SNoR x

L2061

Hypothesis H10 : SNo u

L2062

Hypothesis H11 : x < u

L2063

Hypothesis H12 : SNo v

L2064

Hypothesis H13 : SNo (g u v)

L2065

Hypothesis H14 : x2 ∈ SNoL y

L2066

Hypothesis H15 : y2 ∈ SNoR z

L2067

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2068

Hypothesis H17 : SNo x2

L2069

Hypothesis H18 : x2 < y

L2070

Hypothesis H19 : SNo y2

L2071

Hypothesis H20 : z < y2

L2072

Hypothesis H21 : SNo (g u y)

L2073

Hypothesis H22 : SNo (g u x2)

L2074

Hypothesis H23 : SNo (g x x2)

L2075

Hypothesis H24 : SNo (g x2 z + g y y2)

L2076

Hypothesis H25 : SNo (g x2 y2)

L2077

Theorem. (Conj_mul_SNo_assoc_lem1__58__9)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 88217e... and proposition id is c9dbcf...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__59__2

L2083

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__59__2 TMdsZQFQYzyGLWi3q9pW6XxdjBwwyEXgK5H bounty of about 25 bars ***)

L2084

Variable x : set

L2085

Variable y : set

L2086

Variable z : set

L2087

Variable w : set

L2088

Variable u : set

L2089

Variable v : set

L2090

Variable x2 : set

L2091

Variable y2 : set

L2092

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2093

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2094

Hypothesis H3 : SNo x

L2095

Hypothesis H4 : SNo y

L2096

Hypothesis H5 : SNo z

L2097

Hypothesis H6 : SNo (g x y)

L2098

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2099

Hypothesis H8 : u ∈ SNoR x

L2100

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2101

Hypothesis H10 : SNo u

L2102

Hypothesis H11 : x < u

L2103

Hypothesis H12 : SNo v

L2104

Hypothesis H13 : SNo (g u v)

L2105

Hypothesis H14 : x2 ∈ SNoL y

L2106

Hypothesis H15 : y2 ∈ SNoR z

L2107

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2108

Hypothesis H17 : SNo x2

L2109

Hypothesis H18 : x2 < y

L2110

Hypothesis H19 : SNo y2

L2111

Hypothesis H20 : z < y2

L2112

Hypothesis H21 : SNo (g u y)

L2113

Hypothesis H22 : SNo (g u x2)

L2114

Hypothesis H23 : SNo (g x x2)

L2115

Hypothesis H24 : SNo (g x2 z + g y y2)

L2116

Theorem. (Conj_mul_SNo_assoc_lem1__59__2)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 57f44f... and proposition id is f6aaf0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__59__7

L2122

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__59__7 TMX4a9RBjchAk1uecsvqfCF8noiP8SLCQ47 bounty of about 25 bars ***)

L2123

Variable x : set

L2124

Variable y : set

L2125

Variable z : set

L2126

Variable w : set

L2127

Variable u : set

L2128

Variable v : set

L2129

Variable x2 : set

L2130

Variable y2 : set

L2131

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2132

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2133

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2134

Hypothesis H3 : SNo x

L2135

Hypothesis H4 : SNo y

L2136

Hypothesis H5 : SNo z

L2137

Hypothesis H6 : SNo (g x y)

L2138

Hypothesis H8 : u ∈ SNoR x

L2139

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2140

Hypothesis H10 : SNo u

L2141

Hypothesis H11 : x < u

L2142

Hypothesis H12 : SNo v

L2143

Hypothesis H13 : SNo (g u v)

L2144

Hypothesis H14 : x2 ∈ SNoL y

L2145

Hypothesis H15 : y2 ∈ SNoR z

L2146

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2147

Hypothesis H17 : SNo x2

L2148

Hypothesis H18 : x2 < y

L2149

Hypothesis H19 : SNo y2

L2150

Hypothesis H20 : z < y2

L2151

Hypothesis H21 : SNo (g u y)

L2152

Hypothesis H22 : SNo (g u x2)

L2153

Hypothesis H23 : SNo (g x x2)

L2154

Hypothesis H24 : SNo (g x2 z + g y y2)

L2155

Theorem. (Conj_mul_SNo_assoc_lem1__59__7)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 4b416f... and proposition id is f0df65...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__59__18

L2161

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__59__18 TMKQ72JH5rsStrBMabaJSTKUT9pnKX7kmGG bounty of about 25 bars ***)

L2162

Variable x : set

L2163

Variable y : set

L2164

Variable z : set

L2165

Variable w : set

L2166

Variable u : set

L2167

Variable v : set

L2168

Variable x2 : set

L2169

Variable y2 : set

L2170

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2171

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2172

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2173

Hypothesis H3 : SNo x

L2174

Hypothesis H4 : SNo y

L2175

Hypothesis H5 : SNo z

L2176

Hypothesis H6 : SNo (g x y)

L2177

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2178

Hypothesis H8 : u ∈ SNoR x

L2179

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2180

Hypothesis H10 : SNo u

L2181

Hypothesis H11 : x < u

L2182

Hypothesis H12 : SNo v

L2183

Hypothesis H13 : SNo (g u v)

L2184

Hypothesis H14 : x2 ∈ SNoL y

L2185

Hypothesis H15 : y2 ∈ SNoR z

L2186

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2187

Hypothesis H17 : SNo x2

L2188

Hypothesis H19 : SNo y2

L2189

Hypothesis H20 : z < y2

L2190

Hypothesis H21 : SNo (g u y)

L2191

Hypothesis H22 : SNo (g u x2)

L2192

Hypothesis H23 : SNo (g x x2)

L2193

Hypothesis H24 : SNo (g x2 z + g y y2)

L2194

Theorem. (Conj_mul_SNo_assoc_lem1__59__18)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is c3fad3... and proposition id is dda03d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__60__6

L2200

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__60__6 TMJGoV9fwNVUkCJ6jWKk7y1ySe64VpFa5o1 bounty of about 25 bars ***)

L2201

Variable x : set

L2202

Variable y : set

L2203

Variable z : set

L2204

Variable w : set

L2205

Variable u : set

L2206

Variable v : set

L2207

Variable x2 : set

L2208

Variable y2 : set

L2209

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2210

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2211

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2212

Hypothesis H3 : SNo x

L2213

Hypothesis H4 : SNo y

L2214

Hypothesis H5 : SNo z

L2215

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2216

Hypothesis H8 : u ∈ SNoR x

L2217

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2218

Hypothesis H10 : SNo u

L2219

Hypothesis H11 : x < u

L2220

Hypothesis H12 : SNo v

L2221

Hypothesis H13 : SNo (g u v)

L2222

Hypothesis H14 : x2 ∈ SNoL y

L2223

Hypothesis H15 : y2 ∈ SNoR z

L2224

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2225

Hypothesis H17 : SNo x2

L2226

Hypothesis H18 : x2 < y

L2227

Hypothesis H19 : SNo y2

L2228

Hypothesis H20 : z < y2

L2229

Hypothesis H21 : SNo (g u y)

L2230

Hypothesis H22 : SNo (g u x2)

L2231

Hypothesis H23 : SNo (g x x2)

L2232

Hypothesis H24 : SNo (g x2 z)

L2233

Hypothesis H25 : SNo (g y y2)

L2234

Theorem. (Conj_mul_SNo_assoc_lem1__60__6)

SNo (g x2 z + g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 0407e3... and proposition id is 9faafb...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__61__1

L2240

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__61__1 TMYuMAGqyLDRQwD2RkYaXDx7YU1bV7oYLwS bounty of about 25 bars ***)

L2241

Variable x : set

L2242

Variable y : set

L2243

Variable z : set

L2244

Variable w : set

L2245

Variable u : set

L2246

Variable v : set

L2247

Variable x2 : set

L2248

Variable y2 : set

L2249

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2250

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2251

Hypothesis H3 : SNo x

L2252

Hypothesis H4 : SNo y

L2253

Hypothesis H5 : SNo z

L2254

Hypothesis H6 : SNo (g x y)

L2255

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2256

Hypothesis H8 : u ∈ SNoR x

L2257

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2258

Hypothesis H10 : SNo u

L2259

Hypothesis H11 : x < u

L2260

Hypothesis H12 : SNo v

L2261

Hypothesis H13 : SNo (g u v)

L2262

Hypothesis H14 : x2 ∈ SNoL y

L2263

Hypothesis H15 : y2 ∈ SNoR z

L2264

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2265

Hypothesis H17 : SNo x2

L2266

Hypothesis H18 : x2 < y

L2267

Hypothesis H19 : SNo y2

L2268

Hypothesis H20 : z < y2

L2269

Hypothesis H21 : SNo (g u y)

L2270

Hypothesis H22 : SNo (g u x2)

L2271

Hypothesis H23 : SNo (g x x2)

L2272

Hypothesis H24 : SNo (g x2 z)

L2273

Theorem. (Conj_mul_SNo_assoc_lem1__61__1)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 6db43e... and proposition id is 71aabd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__62__1

L2279

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__62__1 TMdbjiJWGBKVfrYRUnYogyWkoiGrEP2Wy3h bounty of about 25 bars ***)

L2280

Variable x : set

L2281

Variable y : set

L2282

Variable z : set

L2283

Variable w : set

L2284

Variable u : set

L2285

Variable v : set

L2286

Variable x2 : set

L2287

Variable y2 : set

L2288

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2289

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2290

Hypothesis H3 : SNo x

L2291

Hypothesis H4 : SNo y

L2292

Hypothesis H5 : SNo z

L2293

Hypothesis H6 : SNo (g x y)

L2294

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2295

Hypothesis H8 : u ∈ SNoR x

L2296

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2297

Hypothesis H10 : SNo u

L2298

Hypothesis H11 : x < u

L2299

Hypothesis H12 : SNo v

L2300

Hypothesis H13 : SNo (g u v)

L2301

Hypothesis H14 : x2 ∈ SNoL y

L2302

Hypothesis H15 : y2 ∈ SNoR z

L2303

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2304

Hypothesis H17 : SNo x2

L2305

Hypothesis H18 : x2 < y

L2306

Hypothesis H19 : SNo y2

L2307

Hypothesis H20 : z < y2

L2308

Hypothesis H21 : SNo (g u y)

L2309

Hypothesis H22 : SNo (g u x2)

L2310

Hypothesis H23 : SNo (g x x2)

L2311

Theorem. (Conj_mul_SNo_assoc_lem1__62__1)

SNo (g x2 z) → w < g (g x y) z

In Proofgold the corresponding term root is 34b54d... and proposition id is cd6d07...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__64__15

L2317

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__64__15 TMU4qnYzNx3ctpsSEGQUfmNx8BjbpMgRMZw bounty of about 25 bars ***)

L2318

Variable x : set

L2319

Variable y : set

L2320

Variable z : set

L2321

Variable w : set

L2322

Variable u : set

L2323

Variable v : set

L2324

Variable x2 : set

L2325

Variable y2 : set

L2326

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2327

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2328

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2329

Hypothesis H3 : SNo x

L2330

Hypothesis H4 : SNo y

L2331

Hypothesis H5 : SNo z

L2332

Hypothesis H6 : SNo (g x y)

L2333

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2334

Hypothesis H8 : u ∈ SNoR x

L2335

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2336

Hypothesis H10 : SNo u

L2337

Hypothesis H11 : x < u

L2338

Hypothesis H12 : SNo v

L2339

Hypothesis H13 : SNo (g u v)

L2340

Hypothesis H14 : x2 ∈ SNoL y

L2341

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2342

Hypothesis H17 : SNo x2

L2343

Hypothesis H18 : x2 < y

L2344

Hypothesis H19 : SNo y2

L2345

Hypothesis H20 : z < y2

L2346

Hypothesis H21 : SNo (g u y)

L2347

Theorem. (Conj_mul_SNo_assoc_lem1__64__15)

SNo (g u x2) → w < g (g x y) z

In Proofgold the corresponding term root is 57f9d0... and proposition id is dd89cc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__65__9

L2353

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__65__9 TMZmg43T6HkDTgV7q7c1ncG1BgtUzAJFKp4 bounty of about 25 bars ***)

L2354

Variable x : set

L2355

Variable y : set

L2356

Variable z : set

L2357

Variable w : set

L2358

Variable u : set

L2359

Variable v : set

L2360

Variable x2 : set

L2361

Variable y2 : set

L2362

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2363

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2364

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2365

Hypothesis H3 : SNo x

L2366

Hypothesis H4 : SNo y

L2367

Hypothesis H5 : SNo z

L2368

Hypothesis H6 : SNo (g x y)

L2369

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2370

Hypothesis H8 : u ∈ SNoR x

L2371

Hypothesis H10 : SNo u

L2372

Hypothesis H11 : x < u

L2373

Hypothesis H12 : SNo v

L2374

Hypothesis H13 : SNo (g u v)

L2375

Hypothesis H14 : x2 ∈ SNoL y

L2376

Hypothesis H15 : y2 ∈ SNoR z

L2377

Hypothesis H16 : (g x2 z + g y y2) ≤ v + g x2 y2

L2378

Hypothesis H17 : SNo x2

L2379

Hypothesis H18 : x2 < y

L2380

Hypothesis H19 : SNo y2

L2381

Hypothesis H20 : z < y2

L2382

Theorem. (Conj_mul_SNo_assoc_lem1__65__9)

SNo (g u y) → w < g (g x y) z

In Proofgold the corresponding term root is 2fa438... and proposition id is 8160b3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__67__4

L2388

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__67__4 TMd4NQmfKUVFBqSj4WV3Mbr7dhjNuT3GGwD bounty of about 25 bars ***)

L2389

Variable x : set

L2390

Variable y : set

L2391

Variable z : set

L2392

Variable w : set

L2393

Variable u : set

L2394

Variable v : set

L2395

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L2396

Hypothesis H1 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → (∀z2 : set, z2 ∈ SNoR (g x2 y2) → (∀P : prop, (∀w2 : set, w2 ∈ SNoL x2 → (∀u2 : set, u2 ∈ SNoR y2 → (g w2 y2 + g x2 u2) ≤ z2 + g w2 u2 → P)) → (∀w2 : set, w2 ∈ SNoR x2 → (∀u2 : set, u2 ∈ SNoL y2 → (g w2 y2 + g x2 u2) ≤ z2 + g w2 u2 → P)) → P)))

L2397

Hypothesis H2 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L2398

Hypothesis H3 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 ≤ x2 → w2 ≤ y2 → (g z2 y2 + g x2 w2) ≤ g x2 y2 + g z2 w2)

L2399

Hypothesis H5 : SNo y

L2400

Hypothesis H6 : SNo z

L2401

Hypothesis H7 : SNo (g x y)

L2402

Hypothesis H8 : SNo (g y z)

L2403

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, SNo y2 → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → w = g x2 (g y z) + g x y2 + - (g x2 y2) → (g x2 (g z2 z + g y w2) + g x (y2 + g z2 w2)) ≤ g x (g z2 z + g y w2) + g x2 (y2 + g z2 w2) → (g (g x2 y + g x z2) z + g (g x y + g x2 z2) w2) < g (g x y + g x2 z2) z + g (g x2 y + g x z2) w2 → w < g (g x y) z))))

L2404

Hypothesis H10 : u ∈ SNoR x

L2405

Hypothesis H11 : v ∈ SNoR (g y z)

L2406

Hypothesis H12 : w = g u (g y z) + g x v + - (g u v)

L2407

Hypothesis H13 : SNo u

L2408

Hypothesis H14 : x < u

L2409

Theorem. (Conj_mul_SNo_assoc_lem1__67__4)

SNo v → w < g (g x y) z

In Proofgold the corresponding term root is 8613cc... and proposition id is 0fff95...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__70__6

L2415

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__70__6 TMF3YyEvqxapK1tnoU5jE6ShQBJE5gLGMs9 bounty of about 25 bars ***)

L2416

Variable x : set

L2417

Variable y : set

L2418

Variable z : set

L2419

Variable w : set

L2420

Variable u : set

L2421

Variable v : set

L2422

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L2423

Hypothesis H1 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L2424

Hypothesis H2 : SNo x

L2425

Hypothesis H3 : SNo y

L2426

Hypothesis H4 : SNo z

L2427

Hypothesis H5 : SNo (g x y)

L2428

Hypothesis H7 : w < x

L2429

Hypothesis H8 : SNo u

L2430

Hypothesis H9 : y < u

L2431

Hypothesis H10 : SNo v

L2432

Hypothesis H11 : z < v

L2433

Hypothesis H12 : SNo (g w y)

L2434

Hypothesis H13 : SNo (g w u)

L2435

Hypothesis H14 : SNo (g x u)

L2436

Hypothesis H15 : SNo (g w y + g x u)

L2437

Hypothesis H16 : SNo (g x y + g w u)

L2438

Hypothesis H17 : SNo (g (g w y + g x u) z)

L2439

Theorem. (Conj_mul_SNo_assoc_lem1__70__6)

SNo (g (g x y + g w u) z) → (g (g w y + g x u) z + g (g x y + g w u) v) < g (g x y + g w u) z + g (g w y + g x u) v

In Proofgold the corresponding term root is cfa8f6... and proposition id is 740991...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__72__7

L2445

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__72__7 TMVVbpN8Znfv5eVBZcx6X36kdbwHXQWXwSW bounty of about 25 bars ***)

L2446

Variable x : set

L2447

Variable y : set

L2448

Variable z : set

L2449

Variable w : set

L2450

Variable u : set

L2451

Variable v : set

L2452

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L2453

Hypothesis H1 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L2454

Hypothesis H2 : SNo x

L2455

Hypothesis H3 : SNo y

L2456

Hypothesis H4 : SNo z

L2457

Hypothesis H5 : SNo (g x y)

L2458

Hypothesis H6 : SNo w

L2459

Hypothesis H8 : SNo u

L2460

Hypothesis H9 : y < u

L2461

Hypothesis H10 : SNo v

L2462

Hypothesis H11 : z < v

L2463

Hypothesis H12 : SNo (g w y)

L2464

Hypothesis H13 : SNo (g w u)

L2465

Hypothesis H14 : SNo (g x u)

L2466

Hypothesis H15 : SNo (g w y + g x u)

L2467

Theorem. (Conj_mul_SNo_assoc_lem1__72__7)

SNo (g x y + g w u) → (g (g w y + g x u) z + g (g x y + g w u) v) < g (g x y + g w u) z + g (g w y + g x u) v

In Proofgold the corresponding term root is 12a983... and proposition id is 8783fd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__73__3

L2473

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__73__3 TMVctcy7QXVv7RjLNdSMNsBamUoWzgvxfnZ bounty of about 25 bars ***)

L2474

Variable x : set

L2475

Variable y : set

L2476

Variable z : set

L2477

Variable w : set

L2478

Variable u : set

L2479

Variable v : set

L2480

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L2481

Hypothesis H1 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L2482

Hypothesis H2 : SNo x

L2483

Hypothesis H4 : SNo z

L2484

Hypothesis H5 : SNo (g x y)

L2485

Hypothesis H6 : SNo w

L2486

Hypothesis H7 : w < x

L2487

Hypothesis H8 : SNo u

L2488

Hypothesis H9 : y < u

L2489

Hypothesis H10 : SNo v

L2490

Hypothesis H11 : z < v

L2491

Hypothesis H12 : SNo (g w y)

L2492

Hypothesis H13 : SNo (g w u)

L2493

Hypothesis H14 : SNo (g x u)

L2494

Theorem. (Conj_mul_SNo_assoc_lem1__73__3)

SNo (g w y + g x u) → (g (g w y + g x u) z + g (g x y + g w u) v) < g (g x y + g w u) z + g (g w y + g x u) v

In Proofgold the corresponding term root is b2e076... and proposition id is 3632d3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__74__8

L2500

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__74__8 TMWqB49vbQvsj2f7md8HYQMsNnbVKTm2mT7 bounty of about 25 bars ***)

L2501

Variable x : set

L2502

Variable y : set

L2503

Variable z : set

L2504

Variable w : set

L2505

Variable u : set

L2506

Variable v : set

L2507

Variable x2 : set

L2508

Variable y2 : set

L2509

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2510

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2511

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2512

Hypothesis H3 : SNo x

L2513

Hypothesis H4 : SNo y

L2514

Hypothesis H5 : SNo z

L2515

Hypothesis H6 : SNo (g x y)

L2516

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2517

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2518

Hypothesis H10 : SNo u

L2519

Hypothesis H11 : u < x

L2520

Hypothesis H12 : SNo v

L2521

Hypothesis H13 : x2 ∈ SNoR y

L2522

Hypothesis H14 : y2 ∈ SNoR z

L2523

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2524

Hypothesis H16 : SNo x2

L2525

Hypothesis H17 : y < x2

L2526

Hypothesis H18 : SNo y2

L2527

Hypothesis H19 : z < y2

L2528

Hypothesis H20 : SNo (g u y)

L2529

Hypothesis H21 : SNo (g u x2)

L2530

Hypothesis H22 : SNo (g x x2)

L2531

Hypothesis H23 : SNo (g x2 z + g y y2)

L2532

Hypothesis H24 : SNo (g x2 y2)

L2533

Theorem. (Conj_mul_SNo_assoc_lem1__74__8)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is d9ad13... and proposition id is 3d8aec...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__75__0

L2539

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__75__0 TMcYCj9DzpNQET3JEv8NqJ6eCebrA5Jxroo bounty of about 25 bars ***)

L2540

Variable x : set

L2541

Variable y : set

L2542

Variable z : set

L2543

Variable w : set

L2544

Variable u : set

L2545

Variable v : set

L2546

Variable x2 : set

L2547

Variable y2 : set

L2548

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2549

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2550

Hypothesis H3 : SNo x

L2551

Hypothesis H4 : SNo y

L2552

Hypothesis H5 : SNo z

L2553

Hypothesis H6 : SNo (g x y)

L2554

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2555

Hypothesis H8 : u ∈ SNoL x

L2556

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2557

Hypothesis H10 : SNo u

L2558

Hypothesis H11 : u < x

L2559

Hypothesis H12 : SNo v

L2560

Hypothesis H13 : x2 ∈ SNoR y

L2561

Hypothesis H14 : y2 ∈ SNoR z

L2562

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2563

Hypothesis H16 : SNo x2

L2564

Hypothesis H17 : y < x2

L2565

Hypothesis H18 : SNo y2

L2566

Hypothesis H19 : z < y2

L2567

Hypothesis H20 : SNo (g u y)

L2568

Hypothesis H21 : SNo (g u x2)

L2569

Hypothesis H22 : SNo (g x x2)

L2570

Hypothesis H23 : SNo (g x2 z + g y y2)

L2571

Theorem. (Conj_mul_SNo_assoc_lem1__75__0)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is a98402... and proposition id is 581454...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__75__9

L2577

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__75__9 TMaKCqHAaxswiceUCBSMLmVCu4V9dTD5n8b bounty of about 25 bars ***)

L2578

Variable x : set

L2579

Variable y : set

L2580

Variable z : set

L2581

Variable w : set

L2582

Variable u : set

L2583

Variable v : set

L2584

Variable x2 : set

L2585

Variable y2 : set

L2586

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2587

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2588

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2589

Hypothesis H3 : SNo x

L2590

Hypothesis H4 : SNo y

L2591

Hypothesis H5 : SNo z

L2592

Hypothesis H6 : SNo (g x y)

L2593

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2594

Hypothesis H8 : u ∈ SNoL x

L2595

Hypothesis H10 : SNo u

L2596

Hypothesis H11 : u < x

L2597

Hypothesis H12 : SNo v

L2598

Hypothesis H13 : x2 ∈ SNoR y

L2599

Hypothesis H14 : y2 ∈ SNoR z

L2600

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2601

Hypothesis H16 : SNo x2

L2602

Hypothesis H17 : y < x2

L2603

Hypothesis H18 : SNo y2

L2604

Hypothesis H19 : z < y2

L2605

Hypothesis H20 : SNo (g u y)

L2606

Hypothesis H21 : SNo (g u x2)

L2607

Hypothesis H22 : SNo (g x x2)

L2608

Hypothesis H23 : SNo (g x2 z + g y y2)

L2609

Theorem. (Conj_mul_SNo_assoc_lem1__75__9)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 5f487d... and proposition id is 86d7c6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__76__10

L2615

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__76__10 TMcJ22hn8BFMdLtvmvJUJjh6tiv21P6hMu4 bounty of about 25 bars ***)

L2616

Variable x : set

L2617

Variable y : set

L2618

Variable z : set

L2619

Variable w : set

L2620

Variable u : set

L2621

Variable v : set

L2622

Variable x2 : set

L2623

Variable y2 : set

L2624

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2625

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2626

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2627

Hypothesis H3 : SNo x

L2628

Hypothesis H4 : SNo y

L2629

Hypothesis H5 : SNo z

L2630

Hypothesis H6 : SNo (g x y)

L2631

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2632

Hypothesis H8 : u ∈ SNoL x

L2633

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2634

Hypothesis H11 : u < x

L2635

Hypothesis H12 : SNo v

L2636

Hypothesis H13 : x2 ∈ SNoR y

L2637

Hypothesis H14 : y2 ∈ SNoR z

L2638

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2639

Hypothesis H16 : SNo x2

L2640

Hypothesis H17 : y < x2

L2641

Hypothesis H18 : SNo y2

L2642

Hypothesis H19 : z < y2

L2643

Hypothesis H20 : SNo (g u y)

L2644

Hypothesis H21 : SNo (g u x2)

L2645

Hypothesis H22 : SNo (g x x2)

L2646

Hypothesis H23 : SNo (g x2 z)

L2647

Hypothesis H24 : SNo (g y y2)

L2648

Theorem. (Conj_mul_SNo_assoc_lem1__76__10)

SNo (g x2 z + g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 8df412... and proposition id is 4e7fe9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__77__7

L2654

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__77__7 TMVH516dhQQD1Do5jcyRFkVPrUT9pj4qWoL bounty of about 25 bars ***)

L2655

Variable x : set

L2656

Variable y : set

L2657

Variable z : set

L2658

Variable w : set

L2659

Variable u : set

L2660

Variable v : set

L2661

Variable x2 : set

L2662

Variable y2 : set

L2663

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2664

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2665

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2666

Hypothesis H3 : SNo x

L2667

Hypothesis H4 : SNo y

L2668

Hypothesis H5 : SNo z

L2669

Hypothesis H6 : SNo (g x y)

L2670

Hypothesis H8 : u ∈ SNoL x

L2671

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2672

Hypothesis H10 : SNo u

L2673

Hypothesis H11 : u < x

L2674

Hypothesis H12 : SNo v

L2675

Hypothesis H13 : x2 ∈ SNoR y

L2676

Hypothesis H14 : y2 ∈ SNoR z

L2677

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2678

Hypothesis H16 : SNo x2

L2679

Hypothesis H17 : y < x2

L2680

Hypothesis H18 : SNo y2

L2681

Hypothesis H19 : z < y2

L2682

Hypothesis H20 : SNo (g u y)

L2683

Hypothesis H21 : SNo (g u x2)

L2684

Hypothesis H22 : SNo (g x x2)

L2685

Hypothesis H23 : SNo (g x2 z)

L2686

Theorem. (Conj_mul_SNo_assoc_lem1__77__7)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 9fd6f5... and proposition id is 93adb0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__77__13

L2692

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__77__13 TMHSB8oGCmkSCxSv7TEm2tmh9oDM82PRcAD bounty of about 25 bars ***)

L2693

Variable x : set

L2694

Variable y : set

L2695

Variable z : set

L2696

Variable w : set

L2697

Variable u : set

L2698

Variable v : set

L2699

Variable x2 : set

L2700

Variable y2 : set

L2701

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2702

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2703

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2704

Hypothesis H3 : SNo x

L2705

Hypothesis H4 : SNo y

L2706

Hypothesis H5 : SNo z

L2707

Hypothesis H6 : SNo (g x y)

L2708

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2709

Hypothesis H8 : u ∈ SNoL x

L2710

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2711

Hypothesis H10 : SNo u

L2712

Hypothesis H11 : u < x

L2713

Hypothesis H12 : SNo v

L2714

Hypothesis H14 : y2 ∈ SNoR z

L2715

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2716

Hypothesis H16 : SNo x2

L2717

Hypothesis H17 : y < x2

L2718

Hypothesis H18 : SNo y2

L2719

Hypothesis H19 : z < y2

L2720

Hypothesis H20 : SNo (g u y)

L2721

Hypothesis H21 : SNo (g u x2)

L2722

Hypothesis H22 : SNo (g x x2)

L2723

Hypothesis H23 : SNo (g x2 z)

L2724

Theorem. (Conj_mul_SNo_assoc_lem1__77__13)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 54c247... and proposition id is 19caf4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__77__19

L2730

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__77__19 TMUUyTNqprRnkiseku7ZpRcxgMbqB1VY2Jm bounty of about 25 bars ***)

L2731

Variable x : set

L2732

Variable y : set

L2733

Variable z : set

L2734

Variable w : set

L2735

Variable u : set

L2736

Variable v : set

L2737

Variable x2 : set

L2738

Variable y2 : set

L2739

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2740

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2741

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2742

Hypothesis H3 : SNo x

L2743

Hypothesis H4 : SNo y

L2744

Hypothesis H5 : SNo z

L2745

Hypothesis H6 : SNo (g x y)

L2746

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2747

Hypothesis H8 : u ∈ SNoL x

L2748

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2749

Hypothesis H10 : SNo u

L2750

Hypothesis H11 : u < x

L2751

Hypothesis H12 : SNo v

L2752

Hypothesis H13 : x2 ∈ SNoR y

L2753

Hypothesis H14 : y2 ∈ SNoR z

L2754

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2755

Hypothesis H16 : SNo x2

L2756

Hypothesis H17 : y < x2

L2757

Hypothesis H18 : SNo y2

L2758

Hypothesis H20 : SNo (g u y)

L2759

Hypothesis H21 : SNo (g u x2)

L2760

Hypothesis H22 : SNo (g x x2)

L2761

Hypothesis H23 : SNo (g x2 z)

L2762

Theorem. (Conj_mul_SNo_assoc_lem1__77__19)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 3d4667... and proposition id is 6af766...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__79__14

L2768

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__79__14 TMKwh8VuBBQKsdxR4re3WCZgANyJeXaFEoT bounty of about 25 bars ***)

L2769

Variable x : set

L2770

Variable y : set

L2771

Variable z : set

L2772

Variable w : set

L2773

Variable u : set

L2774

Variable v : set

L2775

Variable x2 : set

L2776

Variable y2 : set

L2777

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2778

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2779

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2780

Hypothesis H3 : SNo x

L2781

Hypothesis H4 : SNo y

L2782

Hypothesis H5 : SNo z

L2783

Hypothesis H6 : SNo (g x y)

L2784

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2785

Hypothesis H8 : u ∈ SNoL x

L2786

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2787

Hypothesis H10 : SNo u

L2788

Hypothesis H11 : u < x

L2789

Hypothesis H12 : SNo v

L2790

Hypothesis H13 : x2 ∈ SNoR y

L2791

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2792

Hypothesis H16 : SNo x2

L2793

Hypothesis H17 : y < x2

L2794

Hypothesis H18 : SNo y2

L2795

Hypothesis H19 : z < y2

L2796

Hypothesis H20 : SNo (g u y)

L2797

Hypothesis H21 : SNo (g u x2)

L2798

Theorem. (Conj_mul_SNo_assoc_lem1__79__14)

SNo (g x x2) → w < g (g x y) z

In Proofgold the corresponding term root is a6ddf5... and proposition id is 94c5b7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__80__7

L2804

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__80__7 TMPF9zKzRmofAZYWAWTMTVb8wAio5BtHsrn bounty of about 25 bars ***)

L2805

Variable x : set

L2806

Variable y : set

L2807

Variable z : set

L2808

Variable w : set

L2809

Variable u : set

L2810

Variable v : set

L2811

Variable x2 : set

L2812

Variable y2 : set

L2813

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2814

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2815

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2816

Hypothesis H3 : SNo x

L2817

Hypothesis H4 : SNo y

L2818

Hypothesis H5 : SNo z

L2819

Hypothesis H6 : SNo (g x y)

L2820

Hypothesis H8 : u ∈ SNoL x

L2821

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2822

Hypothesis H10 : SNo u

L2823

Hypothesis H11 : u < x

L2824

Hypothesis H12 : SNo v

L2825

Hypothesis H13 : x2 ∈ SNoR y

L2826

Hypothesis H14 : y2 ∈ SNoR z

L2827

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2828

Hypothesis H16 : SNo x2

L2829

Hypothesis H17 : y < x2

L2830

Hypothesis H18 : SNo y2

L2831

Hypothesis H19 : z < y2

L2832

Hypothesis H20 : SNo (g u y)

L2833

Theorem. (Conj_mul_SNo_assoc_lem1__80__7)

SNo (g u x2) → w < g (g x y) z

In Proofgold the corresponding term root is d8d9da... and proposition id is 315b19...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__81__0

L2839

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__81__0 TMFqFQLd6tEodASQJUpQ2DtkpzFXn63A9Jf bounty of about 25 bars ***)

L2840

Variable x : set

L2841

Variable y : set

L2842

Variable z : set

L2843

Variable w : set

L2844

Variable u : set

L2845

Variable v : set

L2846

Variable x2 : set

L2847

Variable y2 : set

L2848

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2849

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2850

Hypothesis H3 : SNo x

L2851

Hypothesis H4 : SNo y

L2852

Hypothesis H5 : SNo z

L2853

Hypothesis H6 : SNo (g x y)

L2854

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2855

Hypothesis H8 : u ∈ SNoL x

L2856

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2857

Hypothesis H10 : SNo u

L2858

Hypothesis H11 : u < x

L2859

Hypothesis H12 : SNo v

L2860

Hypothesis H13 : x2 ∈ SNoR y

L2861

Hypothesis H14 : y2 ∈ SNoR z

L2862

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2863

Hypothesis H16 : SNo x2

L2864

Hypothesis H17 : y < x2

L2865

Hypothesis H18 : SNo y2

L2866

Hypothesis H19 : z < y2

L2867

Theorem. (Conj_mul_SNo_assoc_lem1__81__0)

SNo (g u y) → w < g (g x y) z

In Proofgold the corresponding term root is 5850a3... and proposition id is 4d5ea7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__82__1

L2873

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__82__1 TMbY5TmQQ2mQK2Ew57sgVa7Ayfxqy1xhJn2 bounty of about 25 bars ***)

L2874

Variable x : set

L2875

Variable y : set

L2876

Variable z : set

L2877

Variable w : set

L2878

Variable u : set

L2879

Variable v : set

L2880

Variable x2 : set

L2881

Variable y2 : set

L2882

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2883

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2884

Hypothesis H3 : SNo x

L2885

Hypothesis H4 : SNo y

L2886

Hypothesis H5 : SNo z

L2887

Hypothesis H6 : SNo (g x y)

L2888

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2889

Hypothesis H8 : u ∈ SNoL x

L2890

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2891

Hypothesis H10 : SNo u

L2892

Hypothesis H11 : u < x

L2893

Hypothesis H12 : SNo v

L2894

Hypothesis H13 : x2 ∈ SNoL y

L2895

Hypothesis H14 : y2 ∈ SNoL z

L2896

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2897

Hypothesis H16 : SNo x2

L2898

Hypothesis H17 : x2 < y

L2899

Hypothesis H18 : SNo y2

L2900

Hypothesis H19 : y2 < z

L2901

Hypothesis H20 : SNo (g x2 z + g y y2)

L2902

Hypothesis H21 : SNo (g x2 y2)

L2903

Theorem. (Conj_mul_SNo_assoc_lem1__82__1)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 9ca4bb... and proposition id is 6db841...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__82__14

L2909

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__82__14 TMMyipcVyjQYKpAW4AvtC1p5qLhemnfNxc8 bounty of about 25 bars ***)

L2910

Variable x : set

L2911

Variable y : set

L2912

Variable z : set

L2913

Variable w : set

L2914

Variable u : set

L2915

Variable v : set

L2916

Variable x2 : set

L2917

Variable y2 : set

L2918

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2919

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2920

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2921

Hypothesis H3 : SNo x

L2922

Hypothesis H4 : SNo y

L2923

Hypothesis H5 : SNo z

L2924

Hypothesis H6 : SNo (g x y)

L2925

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2926

Hypothesis H8 : u ∈ SNoL x

L2927

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2928

Hypothesis H10 : SNo u

L2929

Hypothesis H11 : u < x

L2930

Hypothesis H12 : SNo v

L2931

Hypothesis H13 : x2 ∈ SNoL y

L2932

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2933

Hypothesis H16 : SNo x2

L2934

Hypothesis H17 : x2 < y

L2935

Hypothesis H18 : SNo y2

L2936

Hypothesis H19 : y2 < z

L2937

Hypothesis H20 : SNo (g x2 z + g y y2)

L2938

Hypothesis H21 : SNo (g x2 y2)

L2939

Theorem. (Conj_mul_SNo_assoc_lem1__82__14)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is cd4402... and proposition id is db3818...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__83__9

L2945

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__83__9 TMF9BEKoUTvRvTUhwS99wDwGnqjCzMYa9dS bounty of about 25 bars ***)

L2946

Variable x : set

L2947

Variable y : set

L2948

Variable z : set

L2949

Variable w : set

L2950

Variable u : set

L2951

Variable v : set

L2952

Variable x2 : set

L2953

Variable y2 : set

L2954

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2955

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2956

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2957

Hypothesis H3 : SNo x

L2958

Hypothesis H4 : SNo y

L2959

Hypothesis H5 : SNo z

L2960

Hypothesis H6 : SNo (g x y)

L2961

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2962

Hypothesis H8 : u ∈ SNoL x

L2963

Hypothesis H10 : SNo u

L2964

Hypothesis H11 : u < x

L2965

Hypothesis H12 : SNo v

L2966

Hypothesis H13 : x2 ∈ SNoL y

L2967

Hypothesis H14 : y2 ∈ SNoL z

L2968

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L2969

Hypothesis H16 : SNo x2

L2970

Hypothesis H17 : x2 < y

L2971

Hypothesis H18 : SNo y2

L2972

Hypothesis H19 : y2 < z

L2973

Hypothesis H20 : SNo (g x2 z + g y y2)

L2974

Theorem. (Conj_mul_SNo_assoc_lem1__83__9)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is e18f40... and proposition id is e9f6b8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__83__13

L2980

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__83__13 TMXvHNrm6F4MRxLAdhrrDLnWUVxKdch4csz bounty of about 25 bars ***)

L2981

Variable x : set

L2982

Variable y : set

L2983

Variable z : set

L2984

Variable w : set

L2985

Variable u : set

L2986

Variable v : set

L2987

Variable x2 : set

L2988

Variable y2 : set

L2989

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L2990

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L2991

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L2992

Hypothesis H3 : SNo x

L2993

Hypothesis H4 : SNo y

L2994

Hypothesis H5 : SNo z

L2995

Hypothesis H6 : SNo (g x y)

L2996

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L2997

Hypothesis H8 : u ∈ SNoL x

L2998

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L2999

Hypothesis H10 : SNo u

L3000

Hypothesis H11 : u < x

L3001

Hypothesis H12 : SNo v

L3002

Hypothesis H14 : y2 ∈ SNoL z

L3003

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L3004

Hypothesis H16 : SNo x2

L3005

Hypothesis H17 : x2 < y

L3006

Hypothesis H18 : SNo y2

L3007

Hypothesis H19 : y2 < z

L3008

Hypothesis H20 : SNo (g x2 z + g y y2)

L3009

Theorem. (Conj_mul_SNo_assoc_lem1__83__13)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 548fa5... and proposition id is 278285...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__84__13

L3015

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__84__13 TMX9UtZdV3QoGbnPiYnY3yEudYoyrwQoE7B bounty of about 25 bars ***)

L3016

Variable x : set

L3017

Variable y : set

L3018

Variable z : set

L3019

Variable w : set

L3020

Variable u : set

L3021

Variable v : set

L3022

Variable x2 : set

L3023

Variable y2 : set

L3024

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3025

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L3026

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L3027

Hypothesis H3 : SNo x

L3028

Hypothesis H4 : SNo y

L3029

Hypothesis H5 : SNo z

L3030

Hypothesis H6 : SNo (g x y)

L3031

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L3032

Hypothesis H8 : u ∈ SNoL x

L3033

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L3034

Hypothesis H10 : SNo u

L3035

Hypothesis H11 : u < x

L3036

Hypothesis H12 : SNo v

L3037

Hypothesis H14 : y2 ∈ SNoL z

L3038

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L3039

Hypothesis H16 : SNo x2

L3040

Hypothesis H17 : x2 < y

L3041

Hypothesis H18 : SNo y2

L3042

Hypothesis H19 : y2 < z

L3043

Hypothesis H20 : SNo (g x2 z)

L3044

Hypothesis H21 : SNo (g y y2)

L3045

Theorem. (Conj_mul_SNo_assoc_lem1__84__13)

SNo (g x2 z + g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 2617ed... and proposition id is 7aafa3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__84__20

L3051

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__84__20 TMTyrKic9auo6BaW2S2irVcxxfCBngdwxQA bounty of about 25 bars ***)

L3052

Variable x : set

L3053

Variable y : set

L3054

Variable z : set

L3055

Variable w : set

L3056

Variable u : set

L3057

Variable v : set

L3058

Variable x2 : set

L3059

Variable y2 : set

L3060

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3061

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L3062

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L3063

Hypothesis H3 : SNo x

L3064

Hypothesis H4 : SNo y

L3065

Hypothesis H5 : SNo z

L3066

Hypothesis H6 : SNo (g x y)

L3067

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L3068

Hypothesis H8 : u ∈ SNoL x

L3069

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L3070

Hypothesis H10 : SNo u

L3071

Hypothesis H11 : u < x

L3072

Hypothesis H12 : SNo v

L3073

Hypothesis H13 : x2 ∈ SNoL y

L3074

Hypothesis H14 : y2 ∈ SNoL z

L3075

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L3076

Hypothesis H16 : SNo x2

L3077

Hypothesis H17 : x2 < y

L3078

Hypothesis H18 : SNo y2

L3079

Hypothesis H19 : y2 < z

L3080

Hypothesis H21 : SNo (g y y2)

L3081

Theorem. (Conj_mul_SNo_assoc_lem1__84__20)

SNo (g x2 z + g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 1fb660... and proposition id is ebb2c7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__85__0

L3087

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__85__0 TMcXh6VKA18L7nFmbZPEPVNSkVSmVoZ1pC6 bounty of about 25 bars ***)

L3088

Variable x : set

L3089

Variable y : set

L3090

Variable z : set

L3091

Variable w : set

L3092

Variable u : set

L3093

Variable v : set

L3094

Variable x2 : set

L3095

Variable y2 : set

L3096

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L3097

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L3098

Hypothesis H3 : SNo x

L3099

Hypothesis H4 : SNo y

L3100

Hypothesis H5 : SNo z

L3101

Hypothesis H6 : SNo (g x y)

L3102

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L3103

Hypothesis H8 : u ∈ SNoL x

L3104

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L3105

Hypothesis H10 : SNo u

L3106

Hypothesis H11 : u < x

L3107

Hypothesis H12 : SNo v

L3108

Hypothesis H13 : x2 ∈ SNoL y

L3109

Hypothesis H14 : y2 ∈ SNoL z

L3110

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L3111

Hypothesis H16 : SNo x2

L3112

Hypothesis H17 : x2 < y

L3113

Hypothesis H18 : SNo y2

L3114

Hypothesis H19 : y2 < z

L3115

Hypothesis H20 : SNo (g x2 z)

L3116

Theorem. (Conj_mul_SNo_assoc_lem1__85__0)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is c59a01... and proposition id is 327c35...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__85__3

L3122

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__85__3 TMGfu3w7GguSuHAD8JKrBVt5bodpAqWCxh6 bounty of about 25 bars ***)

L3123

Variable x : set

L3124

Variable y : set

L3125

Variable z : set

L3126

Variable w : set

L3127

Variable u : set

L3128

Variable v : set

L3129

Variable x2 : set

L3130

Variable y2 : set

L3131

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3132

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L3133

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L3134

Hypothesis H4 : SNo y

L3135

Hypothesis H5 : SNo z

L3136

Hypothesis H6 : SNo (g x y)

L3137

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) ≤ g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) → (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 → w < g (g x y) z))))

L3138

Hypothesis H8 : u ∈ SNoL x

L3139

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L3140

Hypothesis H10 : SNo u

L3141

Hypothesis H11 : u < x

L3142

Hypothesis H12 : SNo v

L3143

Hypothesis H13 : x2 ∈ SNoL y

L3144

Hypothesis H14 : y2 ∈ SNoL z

L3145

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L3146

Hypothesis H16 : SNo x2

L3147

Hypothesis H17 : x2 < y

L3148

Hypothesis H18 : SNo y2

L3149

Hypothesis H19 : y2 < z

L3150

Hypothesis H20 : SNo (g x2 z)

L3151

Theorem. (Conj_mul_SNo_assoc_lem1__85__3)

SNo (g y y2) → w < g (g x y) z

In Proofgold the corresponding term root is 90abbe... and proposition id is e4cf77...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__87__3

L3157

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__87__3 TMQ4BtqbMNJVoC8EzDTMZY4r26U93EAgLkN bounty of about 25 bars ***)

L3158

Variable x : set

L3159

Variable y : set

L3160

Variable z : set

L3161

Variable w : set

L3162

Variable u : set

L3163

Variable v : set

L3164

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L3165

Hypothesis H1 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → (∀z2 : set, z2 ∈ SNoL (g x2 y2) → (∀P : prop, (∀w2 : set, w2 ∈ SNoL x2 → (∀u2 : set, u2 ∈ SNoL y2 → (z2 + g w2 u2) ≤ g w2 y2 + g x2 u2 → P)) → (∀w2 : set, w2 ∈ SNoR x2 → (∀u2 : set, u2 ∈ SNoR y2 → (z2 + g w2 u2) ≤ g w2 y2 + g x2 u2 → P)) → P)))

L3166

Hypothesis H2 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L3167

Hypothesis H4 : SNo x

L3168

Hypothesis H5 : SNo y

L3169

Hypothesis H6 : SNo z

L3170

Hypothesis H7 : SNo (g x y)

L3171

Hypothesis H8 : SNo (g y z)

L3172

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, SNo y2 → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → w = g x2 (g y z) + g x y2 + - (g x2 y2) → (g x2 (g z2 z + g y w2) + g x (y2 + g z2 w2)) ≤ g x (g z2 z + g y w2) + g x2 (y2 + g z2 w2) → (g (g x2 y + g x z2) z + g (g x y + g x2 z2) w2) < g (g x y + g x2 z2) z + g (g x2 y + g x z2) w2 → w < g (g x y) z))))

L3173

Hypothesis H10 : u ∈ SNoL x

L3174

Hypothesis H11 : v ∈ SNoL (g y z)

L3175

Hypothesis H12 : w = g u (g y z) + g x v + - (g u v)

L3176

Hypothesis H13 : SNo u

L3177

Hypothesis H14 : u < x

L3178

Theorem. (Conj_mul_SNo_assoc_lem1__87__3)

SNo v → w < g (g x y) z

In Proofgold the corresponding term root is 3da978... and proposition id is 0eed71...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__87__9

L3184

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__87__9 TMTYCi7aJmAK4zSSDJy6BAhpHSQDM8ZLWKF bounty of about 25 bars ***)

L3185

Variable x : set

L3186

Variable y : set

L3187

Variable z : set

L3188

Variable w : set

L3189

Variable u : set

L3190

Variable v : set

L3191

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L3192

Hypothesis H1 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → (∀z2 : set, z2 ∈ SNoL (g x2 y2) → (∀P : prop, (∀w2 : set, w2 ∈ SNoL x2 → (∀u2 : set, u2 ∈ SNoL y2 → (z2 + g w2 u2) ≤ g w2 y2 + g x2 u2 → P)) → (∀w2 : set, w2 ∈ SNoR x2 → (∀u2 : set, u2 ∈ SNoR y2 → (z2 + g w2 u2) ≤ g w2 y2 + g x2 u2 → P)) → P)))

L3193

Hypothesis H2 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L3194

Hypothesis H3 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 ≤ x2 → w2 ≤ y2 → (g z2 y2 + g x2 w2) ≤ g x2 y2 + g z2 w2)

L3195

Hypothesis H4 : SNo x

L3196

Hypothesis H5 : SNo y

L3197

Hypothesis H6 : SNo z

L3198

Hypothesis H7 : SNo (g x y)

L3199

Hypothesis H8 : SNo (g y z)

L3200

Hypothesis H10 : u ∈ SNoL x

L3201

Hypothesis H11 : v ∈ SNoL (g y z)

L3202

Hypothesis H12 : w = g u (g y z) + g x v + - (g u v)

L3203

Hypothesis H13 : SNo u

L3204

Hypothesis H14 : u < x

L3205

Theorem. (Conj_mul_SNo_assoc_lem1__87__9)

SNo v → w < g (g x y) z

In Proofgold the corresponding term root is 9d46bf... and proposition id is 488de5...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__88__0

L3211

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__88__0 TMN81AjjFnmX7vQedvtvdPPrptR266tV4y9 bounty of about 25 bars ***)

L3212

Variable x : set

L3213

Variable y : set

L3214

Variable z : set

L3215

Variable w : set

L3216

Variable u : set

L3217

Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)

L3218

Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)

L3219

Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))

L3220

Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))

L3221

Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)

L3222

Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)

L3223

Hypothesis H7 : SNo x

L3224

Hypothesis H8 : SNo y

L3225

Hypothesis H9 : SNo z

L3226

Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)

L3227

Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)

L3228

Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)

L3229

Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))

L3230

Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))

L3231

Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))

L3232

Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))

L3233

Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))

L3234

Hypothesis H18 : u ∈ w

L3235

Hypothesis H19 : SNo (g x y)

L3236

Hypothesis H20 : SNo (g y z)

L3237

Hypothesis H21 : SNo (g (g x y) z)

L3238

Theorem. (Conj_mul_SNo_assoc_lem1__88__0)

(∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, SNo x2 → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → u = g v (g y z) + g x x2 + - (g v x2) → (g v (g y2 z + g y z2) + g x (x2 + g y2 z2)) ≤ g x (g y2 z + g y z2) + g v (x2 + g y2 z2) → (g (g v y + g x y2) z + g (g x y + g v y2) z2) < g (g x y + g v y2) z + g (g v y + g x y2) z2 → u < g (g x y) z)))) → u < g (g x y) z

In Proofgold the corresponding term root is 90711e... and proposition id is 9e7273...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__88__17

L3244

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__88__17 TMPdvfVshkMrPSnNUccB14SnAorP2zSHNof bounty of about 25 bars ***)

L3245

Variable x : set

L3246

Variable y : set

L3247

Variable z : set

L3248

Variable w : set

L3249

Variable u : set

L3250

Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))

L3251

Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)

L3252

Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)

L3253

Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))

L3254

Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))

L3255

Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)

L3256

Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)

L3257

Hypothesis H7 : SNo x

L3258

Hypothesis H8 : SNo y

L3259

Hypothesis H9 : SNo z

L3260

Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)

L3261

Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)

L3262

Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)

L3263

Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))

L3264

Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))

L3265

Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))

L3266

Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))

L3267

Hypothesis H18 : u ∈ w

L3268

Hypothesis H19 : SNo (g x y)

L3269

Hypothesis H20 : SNo (g y z)

L3270

Hypothesis H21 : SNo (g (g x y) z)

L3271

Theorem. (Conj_mul_SNo_assoc_lem1__88__17)

(∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, SNo x2 → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → u = g v (g y z) + g x x2 + - (g v x2) → (g v (g y2 z + g y z2) + g x (x2 + g y2 z2)) ≤ g x (g y2 z + g y z2) + g v (x2 + g y2 z2) → (g (g v y + g x y2) z + g (g x y + g v y2) z2) < g (g x y + g v y2) z + g (g v y + g x y2) z2 → u < g (g x y) z)))) → u < g (g x y) z

In Proofgold the corresponding term root is 0390ba... and proposition id is f1c5cc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__89__7

L3277

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__89__7 TMLtSG3ZqsAVP6Fhe5qtQjVjmiiEYF64pmi bounty of about 25 bars ***)

L3278

Variable x : set

L3279

Variable y : set

L3280

Variable z : set

L3281

Variable w : set

L3282

Variable u : set

L3283

Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))

L3284

Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)

L3285

Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)

L3286

Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))

L3287

Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))

L3288

Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)

L3289

Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)

L3290

Hypothesis H8 : SNo y

L3291

Hypothesis H9 : SNo z

L3292

Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)

L3293

Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)

L3294

Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)

L3295

Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))

L3296

Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))

L3297

Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))

L3298

Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))

L3299

Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))

L3300

Hypothesis H18 : u ∈ w

L3301

Hypothesis H19 : SNo (g x y)

L3302

Hypothesis H20 : SNo (g y z)

L3303

Theorem. (Conj_mul_SNo_assoc_lem1__89__7)

SNo (g (g x y) z) → u < g (g x y) z

In Proofgold the corresponding term root is 848566... and proposition id is cf75d0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__91__1

L3309

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__91__1 TMaWzbxr85bGNhgcJWDLDDPPVoznmgT7iVV bounty of about 25 bars ***)

L3310

Variable x : set

L3311

Variable y : set

L3312

Variable z : set

L3313

Variable w : set

L3314

Variable u : set

L3315

Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))

L3316

Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)

L3317

Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))

L3318

Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))

L3319

Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)

L3320

Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)

L3321

Hypothesis H7 : SNo x

L3322

Hypothesis H8 : SNo y

L3323

Hypothesis H9 : SNo z

L3324

Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)

L3325

Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)

L3326

Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)

L3327

Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))

L3328

Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))

L3329

Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))

L3330

Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))

L3331

Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))

L3332

Hypothesis H18 : u ∈ w

L3333

Theorem. (Conj_mul_SNo_assoc_lem1__91__1)

SNo (g x y) → u < g (g x y) z

In Proofgold the corresponding term root is 85699b... and proposition id is c75ada...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__2__20

L3339

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__2__20 TMSwHCde2s2cGgmYiBwLN4PZgoewLCghYrW bounty of about 25 bars ***)

L3340

Variable x : set

L3341

Variable y : set

L3342

Variable z : set

L3343

Variable w : set

L3344

Variable u : set

L3345

Variable v : set

L3346

Variable x2 : set

L3347

Variable y2 : set

L3348

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3349

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3350

Hypothesis H2 : SNo x

L3351

Hypothesis H3 : SNo z

L3352

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3353

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3354

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3355

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3356

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3357

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3358

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3359

Hypothesis H11 : SNo (g x y)

L3360

Hypothesis H12 : SNo (g (g x y) z)

L3361

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L3362

Hypothesis H14 : SNo v

L3363

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L3364

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L3365

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L3366

Hypothesis H18 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3367

Hypothesis H19 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3368

Hypothesis H21 : SNo y2

L3369

Hypothesis H22 : SNo (g u (g y z))

L3370

Hypothesis H23 : SNo (g x v)

L3371

Hypothesis H24 : SNo (g x x2)

L3372

Hypothesis H25 : SNo (g u v)

L3373

Hypothesis H26 : SNo (g u x2)

L3374

Hypothesis H27 : SNo (g u y)

L3375

Hypothesis H28 : SNo (g x2 z)

L3376

Hypothesis H29 : SNo (g y y2)

L3377

Hypothesis H30 : SNo (g u (g x2 z))

L3378

Hypothesis H31 : SNo (g u (g y y2))

L3379

Hypothesis H32 : SNo (g x2 y2)

L3380

Hypothesis H33 : SNo (g x (g x2 y2))

L3381

Hypothesis H34 : SNo (g x (g x2 z))

L3382

Hypothesis H35 : SNo (g x (g y y2))

L3383

Hypothesis H36 : SNo (g u (g y z) + g x v)

L3384

Hypothesis H37 : SNo (g (g x y) z + g u v)

L3385

Hypothesis H38 : SNo (g u (g x2 y2))

L3386

Hypothesis H39 : SNo (g u (v + g x2 y2))

L3387

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L3388

Hypothesis H41 : SNo (g u v + g u (g x2 y2))

L3389

Theorem. (Conj_mul_SNo_assoc_lem2__2__20)

SNo (g u (g y z) + g x (g x2 z) + g x (g y y2) + g u v + g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 81bcfd... and proposition id is 61faee...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__2__23

L3395

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__2__23 TMG1J753bNykGjs7ZzPeyqmASGa5XSnvmk7 bounty of about 25 bars ***)

L3396

Variable x : set

L3397

Variable y : set

L3398

Variable z : set

L3399

Variable w : set

L3400

Variable u : set

L3401

Variable v : set

L3402

Variable x2 : set

L3403

Variable y2 : set

L3404

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3405

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3406

Hypothesis H2 : SNo x

L3407

Hypothesis H3 : SNo z

L3408

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3409

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3410

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3411

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3412

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3413

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3414

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3415

Hypothesis H11 : SNo (g x y)

L3416

Hypothesis H12 : SNo (g (g x y) z)

L3417

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L3418

Hypothesis H14 : SNo v

L3419

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L3420

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L3421

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L3422

Hypothesis H18 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3423

Hypothesis H19 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3424

Hypothesis H20 : SNo u

L3425

Hypothesis H21 : SNo y2

L3426

Hypothesis H22 : SNo (g u (g y z))

L3427

Hypothesis H24 : SNo (g x x2)

L3428

Hypothesis H25 : SNo (g u v)

L3429

Hypothesis H26 : SNo (g u x2)

L3430

Hypothesis H27 : SNo (g u y)

L3431

Hypothesis H28 : SNo (g x2 z)

L3432

Hypothesis H29 : SNo (g y y2)

L3433

Hypothesis H30 : SNo (g u (g x2 z))

L3434

Hypothesis H31 : SNo (g u (g y y2))

L3435

Hypothesis H32 : SNo (g x2 y2)

L3436

Hypothesis H33 : SNo (g x (g x2 y2))

L3437

Hypothesis H34 : SNo (g x (g x2 z))

L3438

Hypothesis H35 : SNo (g x (g y y2))

L3439

Hypothesis H36 : SNo (g u (g y z) + g x v)

L3440

Hypothesis H37 : SNo (g (g x y) z + g u v)

L3441

Hypothesis H38 : SNo (g u (g x2 y2))

L3442

Hypothesis H39 : SNo (g u (v + g x2 y2))

L3443

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L3444

Hypothesis H41 : SNo (g u v + g u (g x2 y2))

L3445

Theorem. (Conj_mul_SNo_assoc_lem2__2__23)

SNo (g u (g y z) + g x (g x2 z) + g x (g y y2) + g u v + g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 8e8f19... and proposition id is 0e952b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__3__25

L3451

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__3__25 TMP3m1PPxL5FFxzi7p6ZzzGvPiQ93TbKtwk bounty of about 25 bars ***)

L3452

Variable x : set

L3453

Variable y : set

L3454

Variable z : set

L3455

Variable w : set

L3456

Variable u : set

L3457

Variable v : set

L3458

Variable x2 : set

L3459

Variable y2 : set

L3460

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3461

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3462

Hypothesis H2 : SNo x

L3463

Hypothesis H3 : SNo z

L3464

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3465

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3466

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3467

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3468

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3469

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3470

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3471

Hypothesis H11 : SNo (g x y)

L3472

Hypothesis H12 : SNo (g (g x y) z)

L3473

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L3474

Hypothesis H14 : SNo v

L3475

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L3476

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L3477

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L3478

Hypothesis H18 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3479

Hypothesis H19 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3480

Hypothesis H20 : SNo u

L3481

Hypothesis H21 : SNo y2

L3482

Hypothesis H22 : SNo (g u (g y z))

L3483

Hypothesis H23 : SNo (g x v)

L3484

Hypothesis H24 : SNo (g x x2)

L3485

Hypothesis H26 : SNo (g u x2)

L3486

Hypothesis H27 : SNo (g u y)

L3487

Hypothesis H28 : SNo (g x2 z)

L3488

Hypothesis H29 : SNo (g y y2)

L3489

Hypothesis H30 : SNo (g u (g x2 z))

L3490

Hypothesis H31 : SNo (g u (g y y2))

L3491

Hypothesis H32 : SNo (g x2 y2)

L3492

Hypothesis H33 : SNo (g x (g x2 y2))

L3493

Hypothesis H34 : SNo (g x (g x2 z))

L3494

Hypothesis H35 : SNo (g x (g y y2))

L3495

Hypothesis H36 : SNo (g u (g y z) + g x v)

L3496

Hypothesis H37 : SNo (g (g x y) z + g u v)

L3497

Hypothesis H38 : SNo (g u (g x2 y2))

L3498

Hypothesis H39 : SNo (g u (v + g x2 y2))

L3499

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L3500

Theorem. (Conj_mul_SNo_assoc_lem2__3__25)

SNo (g u v + g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 6d3e67... and proposition id is d0494f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__4__11

L3506

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__4__11 TMdgd4Dg6cWbJKBEaS3RDUih6WQjnddSEDL bounty of about 25 bars ***)

L3507

Variable x : set

L3508

Variable y : set

L3509

Variable z : set

L3510

Variable w : set

L3511

Variable u : set

L3512

Variable v : set

L3513

Variable x2 : set

L3514

Variable y2 : set

L3515

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3516

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3517

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3518

Hypothesis H3 : SNo x

L3519

Hypothesis H4 : SNo z

L3520

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3521

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3522

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3523

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3524

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3525

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3526

Hypothesis H12 : SNo (g x y)

L3527

Hypothesis H13 : SNo (g (g x y) z)

L3528

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3529

Hypothesis H15 : SNo v

L3530

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3531

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3532

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3533

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3534

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3535

Hypothesis H21 : SNo u

L3536

Hypothesis H22 : SNo y2

L3537

Hypothesis H23 : SNo (g u (g y z))

L3538

Hypothesis H24 : SNo (g x v)

L3539

Hypothesis H25 : SNo (g x x2)

L3540

Hypothesis H26 : SNo (g u v)

L3541

Hypothesis H27 : SNo (g u x2)

L3542

Hypothesis H28 : SNo (g u y)

L3543

Hypothesis H29 : SNo (g x2 z)

L3544

Hypothesis H30 : SNo (g y y2)

L3545

Hypothesis H31 : SNo (g u (g x2 z))

L3546

Hypothesis H32 : SNo (g u (g y y2))

L3547

Hypothesis H33 : SNo (g x2 y2)

L3548

Hypothesis H34 : SNo (g x (g x2 y2))

L3549

Hypothesis H35 : SNo (g x (g x2 z))

L3550

Hypothesis H36 : SNo (g x (g y y2))

L3551

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3552

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3553

Hypothesis H39 : SNo (g u (g x2 y2))

L3554

Hypothesis H40 : SNo (g u (v + g x2 y2))

L3555

Theorem. (Conj_mul_SNo_assoc_lem2__4__11)

SNo (g u (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 7c0b53... and proposition id is 29969c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__4__14

L3561

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__4__14 TMWDgPg88ywaHDLkv8Ws3GZ2xzGg3ro6oZ9 bounty of about 25 bars ***)

L3562

Variable x : set

L3563

Variable y : set

L3564

Variable z : set

L3565

Variable w : set

L3566

Variable u : set

L3567

Variable v : set

L3568

Variable x2 : set

L3569

Variable y2 : set

L3570

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3571

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3572

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3573

Hypothesis H3 : SNo x

L3574

Hypothesis H4 : SNo z

L3575

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3576

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3577

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3578

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3579

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3580

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3581

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3582

Hypothesis H12 : SNo (g x y)

L3583

Hypothesis H13 : SNo (g (g x y) z)

L3584

Hypothesis H15 : SNo v

L3585

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3586

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3587

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3588

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3589

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3590

Hypothesis H21 : SNo u

L3591

Hypothesis H22 : SNo y2

L3592

Hypothesis H23 : SNo (g u (g y z))

L3593

Hypothesis H24 : SNo (g x v)

L3594

Hypothesis H25 : SNo (g x x2)

L3595

Hypothesis H26 : SNo (g u v)

L3596

Hypothesis H27 : SNo (g u x2)

L3597

Hypothesis H28 : SNo (g u y)

L3598

Hypothesis H29 : SNo (g x2 z)

L3599

Hypothesis H30 : SNo (g y y2)

L3600

Hypothesis H31 : SNo (g u (g x2 z))

L3601

Hypothesis H32 : SNo (g u (g y y2))

L3602

Hypothesis H33 : SNo (g x2 y2)

L3603

Hypothesis H34 : SNo (g x (g x2 y2))

L3604

Hypothesis H35 : SNo (g x (g x2 z))

L3605

Hypothesis H36 : SNo (g x (g y y2))

L3606

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3607

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3608

Hypothesis H39 : SNo (g u (g x2 y2))

L3609

Hypothesis H40 : SNo (g u (v + g x2 y2))

L3610

Theorem. (Conj_mul_SNo_assoc_lem2__4__14)

SNo (g u (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 552c54... and proposition id is 748e48...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__5__6

L3616

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__5__6 TMNJvMdMhPm2esaH1eMT4ywWKugZw7LZz1V bounty of about 25 bars ***)

L3617

Variable x : set

L3618

Variable y : set

L3619

Variable z : set

L3620

Variable w : set

L3621

Variable u : set

L3622

Variable v : set

L3623

Variable x2 : set

L3624

Variable y2 : set

L3625

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3626

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3627

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3628

Hypothesis H3 : SNo x

L3629

Hypothesis H4 : SNo z

L3630

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3631

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3632

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3633

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3634

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3635

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3636

Hypothesis H12 : SNo (g x y)

L3637

Hypothesis H13 : SNo (g (g x y) z)

L3638

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3639

Hypothesis H15 : SNo v

L3640

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3641

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3642

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3643

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3644

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3645

Hypothesis H21 : SNo u

L3646

Hypothesis H22 : SNo y2

L3647

Hypothesis H23 : SNo (g u (g y z))

L3648

Hypothesis H24 : SNo (g x v)

L3649

Hypothesis H25 : SNo (g x x2)

L3650

Hypothesis H26 : SNo (g u v)

L3651

Hypothesis H27 : SNo (g u x2)

L3652

Hypothesis H28 : SNo (g u y)

L3653

Hypothesis H29 : SNo (g x2 z)

L3654

Hypothesis H30 : SNo (g y y2)

L3655

Hypothesis H31 : SNo (g u (g x2 z))

L3656

Hypothesis H32 : SNo (g u (g y y2))

L3657

Hypothesis H33 : SNo (g x2 y2)

L3658

Hypothesis H34 : SNo (g x (g x2 y2))

L3659

Hypothesis H35 : SNo (g x (g x2 z))

L3660

Hypothesis H36 : SNo (g x (g y y2))

L3661

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3662

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3663

Hypothesis H39 : SNo (g u (g x2 y2))

L3664

Hypothesis H40 : SNo (v + g x2 y2)

L3665

Theorem. (Conj_mul_SNo_assoc_lem2__5__6)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 466147... and proposition id is 68574d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__5__26

L3671

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__5__26 TMRHyve96Qn8z611tU37yLXB7v32eGdLgeh bounty of about 25 bars ***)

L3672

Variable x : set

L3673

Variable y : set

L3674

Variable z : set

L3675

Variable w : set

L3676

Variable u : set

L3677

Variable v : set

L3678

Variable x2 : set

L3679

Variable y2 : set

L3680

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3681

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3682

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3683

Hypothesis H3 : SNo x

L3684

Hypothesis H4 : SNo z

L3685

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3686

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3687

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3688

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3689

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3690

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3691

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3692

Hypothesis H12 : SNo (g x y)

L3693

Hypothesis H13 : SNo (g (g x y) z)

L3694

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3695

Hypothesis H15 : SNo v

L3696

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3697

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3698

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3699

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3700

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3701

Hypothesis H21 : SNo u

L3702

Hypothesis H22 : SNo y2

L3703

Hypothesis H23 : SNo (g u (g y z))

L3704

Hypothesis H24 : SNo (g x v)

L3705

Hypothesis H25 : SNo (g x x2)

L3706

Hypothesis H27 : SNo (g u x2)

L3707

Hypothesis H28 : SNo (g u y)

L3708

Hypothesis H29 : SNo (g x2 z)

L3709

Hypothesis H30 : SNo (g y y2)

L3710

Hypothesis H31 : SNo (g u (g x2 z))

L3711

Hypothesis H32 : SNo (g u (g y y2))

L3712

Hypothesis H33 : SNo (g x2 y2)

L3713

Hypothesis H34 : SNo (g x (g x2 y2))

L3714

Hypothesis H35 : SNo (g x (g x2 z))

L3715

Hypothesis H36 : SNo (g x (g y y2))

L3716

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3717

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3718

Hypothesis H39 : SNo (g u (g x2 y2))

L3719

Hypothesis H40 : SNo (v + g x2 y2)

L3720

Theorem. (Conj_mul_SNo_assoc_lem2__5__26)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is f49c69... and proposition id is 690375...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__6__10

L3726

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__6__10 TMPsBVfeCa5SCbL24iYLBgwxxq4ZKWnGW99 bounty of about 25 bars ***)

L3727

Variable x : set

L3728

Variable y : set

L3729

Variable z : set

L3730

Variable w : set

L3731

Variable u : set

L3732

Variable v : set

L3733

Variable x2 : set

L3734

Variable y2 : set

L3735

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3736

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3737

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3738

Hypothesis H3 : SNo x

L3739

Hypothesis H4 : SNo z

L3740

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3741

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3742

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3743

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3744

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3745

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3746

Hypothesis H12 : SNo (g x y)

L3747

Hypothesis H13 : SNo (g (g x y) z)

L3748

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3749

Hypothesis H15 : SNo v

L3750

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3751

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3752

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3753

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3754

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3755

Hypothesis H21 : SNo u

L3756

Hypothesis H22 : SNo y2

L3757

Hypothesis H23 : SNo (g u (g y z))

L3758

Hypothesis H24 : SNo (g x v)

L3759

Hypothesis H25 : SNo (g x x2)

L3760

Hypothesis H26 : SNo (g u v)

L3761

Hypothesis H27 : SNo (g u x2)

L3762

Hypothesis H28 : SNo (g u y)

L3763

Hypothesis H29 : SNo (g x2 z)

L3764

Hypothesis H30 : SNo (g y y2)

L3765

Hypothesis H31 : SNo (g u (g x2 z))

L3766

Hypothesis H32 : SNo (g u (g y y2))

L3767

Hypothesis H33 : SNo (g x2 y2)

L3768

Hypothesis H34 : SNo (g x (g x2 y2))

L3769

Hypothesis H35 : SNo (g x (g x2 z))

L3770

Hypothesis H36 : SNo (g x (g y y2))

L3771

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3772

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3773

Hypothesis H39 : SNo (g u (g x2 y2))

L3774

Theorem. (Conj_mul_SNo_assoc_lem2__6__10)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 83331a... and proposition id is d1dc7c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__6__13

L3780

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__6__13 TMbd3bvgQ5FEve2rcGjxc7aSBpU89432Naj bounty of about 25 bars ***)

L3781

Variable x : set

L3782

Variable y : set

L3783

Variable z : set

L3784

Variable w : set

L3785

Variable u : set

L3786

Variable v : set

L3787

Variable x2 : set

L3788

Variable y2 : set

L3789

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3790

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3791

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3792

Hypothesis H3 : SNo x

L3793

Hypothesis H4 : SNo z

L3794

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3795

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3796

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3797

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3798

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3799

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3800

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3801

Hypothesis H12 : SNo (g x y)

L3802

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3803

Hypothesis H15 : SNo v

L3804

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3805

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3806

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3807

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3808

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3809

Hypothesis H21 : SNo u

L3810

Hypothesis H22 : SNo y2

L3811

Hypothesis H23 : SNo (g u (g y z))

L3812

Hypothesis H24 : SNo (g x v)

L3813

Hypothesis H25 : SNo (g x x2)

L3814

Hypothesis H26 : SNo (g u v)

L3815

Hypothesis H27 : SNo (g u x2)

L3816

Hypothesis H28 : SNo (g u y)

L3817

Hypothesis H29 : SNo (g x2 z)

L3818

Hypothesis H30 : SNo (g y y2)

L3819

Hypothesis H31 : SNo (g u (g x2 z))

L3820

Hypothesis H32 : SNo (g u (g y y2))

L3821

Hypothesis H33 : SNo (g x2 y2)

L3822

Hypothesis H34 : SNo (g x (g x2 y2))

L3823

Hypothesis H35 : SNo (g x (g x2 z))

L3824

Hypothesis H36 : SNo (g x (g y y2))

L3825

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3826

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3827

Hypothesis H39 : SNo (g u (g x2 y2))

L3828

Theorem. (Conj_mul_SNo_assoc_lem2__6__13)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 26c553... and proposition id is a6165c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__6__14

L3834

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__6__14 TMSQasZkSP4JYBpdLYG2EYmtu6sr4CscZ5S bounty of about 25 bars ***)

L3835

Variable x : set

L3836

Variable y : set

L3837

Variable z : set

L3838

Variable w : set

L3839

Variable u : set

L3840

Variable v : set

L3841

Variable x2 : set

L3842

Variable y2 : set

L3843

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3844

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3845

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3846

Hypothesis H3 : SNo x

L3847

Hypothesis H4 : SNo z

L3848

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3849

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3850

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3851

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3852

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3853

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3854

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3855

Hypothesis H12 : SNo (g x y)

L3856

Hypothesis H13 : SNo (g (g x y) z)

L3857

Hypothesis H15 : SNo v

L3858

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3859

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3860

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3861

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3862

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3863

Hypothesis H21 : SNo u

L3864

Hypothesis H22 : SNo y2

L3865

Hypothesis H23 : SNo (g u (g y z))

L3866

Hypothesis H24 : SNo (g x v)

L3867

Hypothesis H25 : SNo (g x x2)

L3868

Hypothesis H26 : SNo (g u v)

L3869

Hypothesis H27 : SNo (g u x2)

L3870

Hypothesis H28 : SNo (g u y)

L3871

Hypothesis H29 : SNo (g x2 z)

L3872

Hypothesis H30 : SNo (g y y2)

L3873

Hypothesis H31 : SNo (g u (g x2 z))

L3874

Hypothesis H32 : SNo (g u (g y y2))

L3875

Hypothesis H33 : SNo (g x2 y2)

L3876

Hypothesis H34 : SNo (g x (g x2 y2))

L3877

Hypothesis H35 : SNo (g x (g x2 z))

L3878

Hypothesis H36 : SNo (g x (g y y2))

L3879

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3880

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3881

Hypothesis H39 : SNo (g u (g x2 y2))

L3882

Theorem. (Conj_mul_SNo_assoc_lem2__6__14)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 3ba18d... and proposition id is a739d4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__6__15

L3888

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__6__15 TMXbWqshkjogDbpga18Dc3oECUhNCLTwkFr bounty of about 25 bars ***)

L3889

Variable x : set

L3890

Variable y : set

L3891

Variable z : set

L3892

Variable w : set

L3893

Variable u : set

L3894

Variable v : set

L3895

Variable x2 : set

L3896

Variable y2 : set

L3897

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3898

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3899

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3900

Hypothesis H3 : SNo x

L3901

Hypothesis H4 : SNo z

L3902

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3903

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3904

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3905

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3906

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3907

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3908

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3909

Hypothesis H12 : SNo (g x y)

L3910

Hypothesis H13 : SNo (g (g x y) z)

L3911

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3912

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3913

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3914

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3915

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3916

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3917

Hypothesis H21 : SNo u

L3918

Hypothesis H22 : SNo y2

L3919

Hypothesis H23 : SNo (g u (g y z))

L3920

Hypothesis H24 : SNo (g x v)

L3921

Hypothesis H25 : SNo (g x x2)

L3922

Hypothesis H26 : SNo (g u v)

L3923

Hypothesis H27 : SNo (g u x2)

L3924

Hypothesis H28 : SNo (g u y)

L3925

Hypothesis H29 : SNo (g x2 z)

L3926

Hypothesis H30 : SNo (g y y2)

L3927

Hypothesis H31 : SNo (g u (g x2 z))

L3928

Hypothesis H32 : SNo (g u (g y y2))

L3929

Hypothesis H33 : SNo (g x2 y2)

L3930

Hypothesis H34 : SNo (g x (g x2 y2))

L3931

Hypothesis H35 : SNo (g x (g x2 z))

L3932

Hypothesis H36 : SNo (g x (g y y2))

L3933

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3934

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3935

Hypothesis H39 : SNo (g u (g x2 y2))

L3936

Theorem. (Conj_mul_SNo_assoc_lem2__6__15)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is b19f82... and proposition id is 199f62...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__6__29

L3942

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__6__29 TMTehYaDEgN7LhqeV8mwysLmgBRBtMugkRq bounty of about 25 bars ***)

L3943

Variable x : set

L3944

Variable y : set

L3945

Variable z : set

L3946

Variable w : set

L3947

Variable u : set

L3948

Variable v : set

L3949

Variable x2 : set

L3950

Variable y2 : set

L3951

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L3952

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L3953

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L3954

Hypothesis H3 : SNo x

L3955

Hypothesis H4 : SNo z

L3956

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L3957

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L3958

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L3959

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L3960

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L3961

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L3962

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L3963

Hypothesis H12 : SNo (g x y)

L3964

Hypothesis H13 : SNo (g (g x y) z)

L3965

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L3966

Hypothesis H15 : SNo v

L3967

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L3968

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L3969

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L3970

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L3971

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L3972

Hypothesis H21 : SNo u

L3973

Hypothesis H22 : SNo y2

L3974

Hypothesis H23 : SNo (g u (g y z))

L3975

Hypothesis H24 : SNo (g x v)

L3976

Hypothesis H25 : SNo (g x x2)

L3977

Hypothesis H26 : SNo (g u v)

L3978

Hypothesis H27 : SNo (g u x2)

L3979

Hypothesis H28 : SNo (g u y)

L3980

Hypothesis H30 : SNo (g y y2)

L3981

Hypothesis H31 : SNo (g u (g x2 z))

L3982

Hypothesis H32 : SNo (g u (g y y2))

L3983

Hypothesis H33 : SNo (g x2 y2)

L3984

Hypothesis H34 : SNo (g x (g x2 y2))

L3985

Hypothesis H35 : SNo (g x (g x2 z))

L3986

Hypothesis H36 : SNo (g x (g y y2))

L3987

Hypothesis H37 : SNo (g u (g y z) + g x v)

L3988

Hypothesis H38 : SNo (g (g x y) z + g u v)

L3989

Hypothesis H39 : SNo (g u (g x2 y2))

L3990

Theorem. (Conj_mul_SNo_assoc_lem2__6__29)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is b220f0... and proposition id is fbc0a7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__6__30

L3996

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__6__30 TMVEzVkwtfqJBG3RDSw4769wqboegLah2b7 bounty of about 25 bars ***)

L3997

Variable x : set

L3998

Variable y : set

L3999

Variable z : set

L4000

Variable w : set

L4001

Variable u : set

L4002

Variable v : set

L4003

Variable x2 : set

L4004

Variable y2 : set

L4005

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4006

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4007

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4008

Hypothesis H3 : SNo x

L4009

Hypothesis H4 : SNo z

L4010

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4011

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4012

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4013

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4014

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4015

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4016

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4017

Hypothesis H12 : SNo (g x y)

L4018

Hypothesis H13 : SNo (g (g x y) z)

L4019

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4020

Hypothesis H15 : SNo v

L4021

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4022

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4023

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4024

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4025

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4026

Hypothesis H21 : SNo u

L4027

Hypothesis H22 : SNo y2

L4028

Hypothesis H23 : SNo (g u (g y z))

L4029

Hypothesis H24 : SNo (g x v)

L4030

Hypothesis H25 : SNo (g x x2)

L4031

Hypothesis H26 : SNo (g u v)

L4032

Hypothesis H27 : SNo (g u x2)

L4033

Hypothesis H28 : SNo (g u y)

L4034

Hypothesis H29 : SNo (g x2 z)

L4035

Hypothesis H31 : SNo (g u (g x2 z))

L4036

Hypothesis H32 : SNo (g u (g y y2))

L4037

Hypothesis H33 : SNo (g x2 y2)

L4038

Hypothesis H34 : SNo (g x (g x2 y2))

L4039

Hypothesis H35 : SNo (g x (g x2 z))

L4040

Hypothesis H36 : SNo (g x (g y y2))

L4041

Hypothesis H37 : SNo (g u (g y z) + g x v)

L4042

Hypothesis H38 : SNo (g (g x y) z + g u v)

L4043

Hypothesis H39 : SNo (g u (g x2 y2))

L4044

Theorem. (Conj_mul_SNo_assoc_lem2__6__30)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is b43c92... and proposition id is 112091...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__7__13

L4050

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__7__13 TMLqkAa6cjByos2FXsvZTDVTpx7egYirX5y bounty of about 25 bars ***)

L4051

Variable x : set

L4052

Variable y : set

L4053

Variable z : set

L4054

Variable w : set

L4055

Variable u : set

L4056

Variable v : set

L4057

Variable x2 : set

L4058

Variable y2 : set

L4059

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4060

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4061

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4062

Hypothesis H3 : SNo x

L4063

Hypothesis H4 : SNo z

L4064

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4065

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4066

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4067

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4068

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4069

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4070

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4071

Hypothesis H12 : SNo (g x y)

L4072

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4073

Hypothesis H15 : SNo v

L4074

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4075

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4076

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4077

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4078

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4079

Hypothesis H21 : SNo u

L4080

Hypothesis H22 : SNo y2

L4081

Hypothesis H23 : SNo (g u (g y z))

L4082

Hypothesis H24 : SNo (g x v)

L4083

Hypothesis H25 : SNo (g x x2)

L4084

Hypothesis H26 : SNo (g u v)

L4085

Hypothesis H27 : SNo (g u x2)

L4086

Hypothesis H28 : SNo (g u y)

L4087

Hypothesis H29 : SNo (g x2 z)

L4088

Hypothesis H30 : SNo (g y y2)

L4089

Hypothesis H31 : SNo (g u (g x2 z))

L4090

Hypothesis H32 : SNo (g u (g y y2))

L4091

Hypothesis H33 : SNo (g x2 y2)

L4092

Hypothesis H34 : SNo (g x (g x2 y2))

L4093

Hypothesis H35 : SNo (g x (g x2 z))

L4094

Hypothesis H36 : SNo (g x (g y y2))

L4095

Hypothesis H37 : SNo (g u (g y z) + g x v)

L4096

Hypothesis H38 : SNo (g (g x y) z + g u v)

L4097

Theorem. (Conj_mul_SNo_assoc_lem2__7__13)

SNo (g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 8f3907... and proposition id is 2d512c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__7__16

L4103

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__7__16 TMKuedjg9Q81PeJnogSNmjtaPvGjX4P2AN3 bounty of about 25 bars ***)

L4104

Variable x : set

L4105

Variable y : set

L4106

Variable z : set

L4107

Variable w : set

L4108

Variable u : set

L4109

Variable v : set

L4110

Variable x2 : set

L4111

Variable y2 : set

L4112

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4113

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4114

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4115

Hypothesis H3 : SNo x

L4116

Hypothesis H4 : SNo z

L4117

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4118

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4119

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4120

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4121

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4122

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4123

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4124

Hypothesis H12 : SNo (g x y)

L4125

Hypothesis H13 : SNo (g (g x y) z)

L4126

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4127

Hypothesis H15 : SNo v

L4128

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4129

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4130

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4131

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4132

Hypothesis H21 : SNo u

L4133

Hypothesis H22 : SNo y2

L4134

Hypothesis H23 : SNo (g u (g y z))

L4135

Hypothesis H24 : SNo (g x v)

L4136

Hypothesis H25 : SNo (g x x2)

L4137

Hypothesis H26 : SNo (g u v)

L4138

Hypothesis H27 : SNo (g u x2)

L4139

Hypothesis H28 : SNo (g u y)

L4140

Hypothesis H29 : SNo (g x2 z)

L4141

Hypothesis H30 : SNo (g y y2)

L4142

Hypothesis H31 : SNo (g u (g x2 z))

L4143

Hypothesis H32 : SNo (g u (g y y2))

L4144

Hypothesis H33 : SNo (g x2 y2)

L4145

Hypothesis H34 : SNo (g x (g x2 y2))

L4146

Hypothesis H35 : SNo (g x (g x2 z))

L4147

Hypothesis H36 : SNo (g x (g y y2))

L4148

Hypothesis H37 : SNo (g u (g y z) + g x v)

L4149

Hypothesis H38 : SNo (g (g x y) z + g u v)

L4150

Theorem. (Conj_mul_SNo_assoc_lem2__7__16)

SNo (g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 19c314... and proposition id is ed00db...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__8__2

L4156

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__8__2 TMav2ZHtL5RVmHpP7g2smWHsXiqdvQn4W3J bounty of about 25 bars ***)

L4157

Variable x : set

L4158

Variable y : set

L4159

Variable z : set

L4160

Variable w : set

L4161

Variable u : set

L4162

Variable v : set

L4163

Variable x2 : set

L4164

Variable y2 : set

L4165

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4166

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4167

Hypothesis H3 : SNo x

L4168

Hypothesis H4 : SNo z

L4169

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4170

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4171

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4172

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4173

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4174

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4175

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4176

Hypothesis H12 : SNo (g x y)

L4177

Hypothesis H13 : SNo (g (g x y) z)

L4178

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4179

Hypothesis H15 : SNo v

L4180

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4181

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4182

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4183

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4184

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4185

Hypothesis H21 : SNo u

L4186

Hypothesis H22 : SNo y2

L4187

Hypothesis H23 : SNo (g u (g y z))

L4188

Hypothesis H24 : SNo (g x v)

L4189

Hypothesis H25 : SNo (g x x2)

L4190

Hypothesis H26 : SNo (g u v)

L4191

Hypothesis H27 : SNo (g u x2)

L4192

Hypothesis H28 : SNo (g u y)

L4193

Hypothesis H29 : SNo (g x2 z)

L4194

Hypothesis H30 : SNo (g y y2)

L4195

Hypothesis H31 : SNo (g u (g x2 z))

L4196

Hypothesis H32 : SNo (g u (g y y2))

L4197

Hypothesis H33 : SNo (g x2 y2)

L4198

Hypothesis H34 : SNo (g x (g x2 y2))

L4199

Hypothesis H35 : SNo (g x (g x2 z))

L4200

Hypothesis H36 : SNo (g x (g y y2))

L4201

Hypothesis H37 : SNo (g u (g y z) + g x v)

L4202

Theorem. (Conj_mul_SNo_assoc_lem2__8__2)

SNo (g (g x y) z + g u v) → g (g x y) z < w

In Proofgold the corresponding term root is ae82fa... and proposition id is c63192...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__8__6

L4208

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__8__6 TMJSUSkeGyWWnMyy3FybfAzpUU6BhwGn3Zb bounty of about 25 bars ***)

L4209

Variable x : set

L4210

Variable y : set

L4211

Variable z : set

L4212

Variable w : set

L4213

Variable u : set

L4214

Variable v : set

L4215

Variable x2 : set

L4216

Variable y2 : set

L4217

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4218

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4219

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4220

Hypothesis H3 : SNo x

L4221

Hypothesis H4 : SNo z

L4222

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4223

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4224

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4225

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4226

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4227

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4228

Hypothesis H12 : SNo (g x y)

L4229

Hypothesis H13 : SNo (g (g x y) z)

L4230

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4231

Hypothesis H15 : SNo v

L4232

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4233

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4234

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4235

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4236

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4237

Hypothesis H21 : SNo u

L4238

Hypothesis H22 : SNo y2

L4239

Hypothesis H23 : SNo (g u (g y z))

L4240

Hypothesis H24 : SNo (g x v)

L4241

Hypothesis H25 : SNo (g x x2)

L4242

Hypothesis H26 : SNo (g u v)

L4243

Hypothesis H27 : SNo (g u x2)

L4244

Hypothesis H28 : SNo (g u y)

L4245

Hypothesis H29 : SNo (g x2 z)

L4246

Hypothesis H30 : SNo (g y y2)

L4247

Hypothesis H31 : SNo (g u (g x2 z))

L4248

Hypothesis H32 : SNo (g u (g y y2))

L4249

Hypothesis H33 : SNo (g x2 y2)

L4250

Hypothesis H34 : SNo (g x (g x2 y2))

L4251

Hypothesis H35 : SNo (g x (g x2 z))

L4252

Hypothesis H36 : SNo (g x (g y y2))

L4253

Hypothesis H37 : SNo (g u (g y z) + g x v)

L4254

Theorem. (Conj_mul_SNo_assoc_lem2__8__6)

SNo (g (g x y) z + g u v) → g (g x y) z < w

In Proofgold the corresponding term root is 157d85... and proposition id is a1fa24...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__9__6

L4260

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__9__6 TMaV2waWALe6Zytp4XSayWHxSDNLe8z7xme bounty of about 25 bars ***)

L4261

Variable x : set

L4262

Variable y : set

L4263

Variable z : set

L4264

Variable w : set

L4265

Variable u : set

L4266

Variable v : set

L4267

Variable x2 : set

L4268

Variable y2 : set

L4269

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4270

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4271

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4272

Hypothesis H3 : SNo x

L4273

Hypothesis H4 : SNo z

L4274

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4275

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4276

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4277

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4278

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4279

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4280

Hypothesis H12 : SNo (g x y)

L4281

Hypothesis H13 : SNo (g (g x y) z)

L4282

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4283

Hypothesis H15 : SNo v

L4284

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4285

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4286

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4287

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4288

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4289

Hypothesis H21 : SNo u

L4290

Hypothesis H22 : SNo y2

L4291

Hypothesis H23 : SNo (g u (g y z))

L4292

Hypothesis H24 : SNo (g x v)

L4293

Hypothesis H25 : SNo (g x x2)

L4294

Hypothesis H26 : SNo (g u v)

L4295

Hypothesis H27 : SNo (g u x2)

L4296

Hypothesis H28 : SNo (g u y)

L4297

Hypothesis H29 : SNo (g x2 z)

L4298

Hypothesis H30 : SNo (g y y2)

L4299

Hypothesis H31 : SNo (g u (g x2 z))

L4300

Hypothesis H32 : SNo (g u (g y y2))

L4301

Hypothesis H33 : SNo (g x2 y2)

L4302

Hypothesis H34 : SNo (g x (g x2 y2))

L4303

Hypothesis H35 : SNo (g x (g x2 z))

L4304

Hypothesis H36 : SNo (g x (g y y2))

L4305

Theorem. (Conj_mul_SNo_assoc_lem2__9__6)

SNo (g u (g y z) + g x v) → g (g x y) z < w

In Proofgold the corresponding term root is 10df90... and proposition id is 484e31...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__9__21

L4311

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__9__21 TMT4SdZqdAMZx6BKPuLhvToyKHvhu1eJsDD bounty of about 25 bars ***)

L4312

Variable x : set

L4313

Variable y : set

L4314

Variable z : set

L4315

Variable w : set

L4316

Variable u : set

L4317

Variable v : set

L4318

Variable x2 : set

L4319

Variable y2 : set

L4320

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4321

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4322

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4323

Hypothesis H3 : SNo x

L4324

Hypothesis H4 : SNo z

L4325

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4326

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4327

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4328

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4329

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4330

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4331

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4332

Hypothesis H12 : SNo (g x y)

L4333

Hypothesis H13 : SNo (g (g x y) z)

L4334

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4335

Hypothesis H15 : SNo v

L4336

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4337

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4338

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4339

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4340

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4341

Hypothesis H22 : SNo y2

L4342

Hypothesis H23 : SNo (g u (g y z))

L4343

Hypothesis H24 : SNo (g x v)

L4344

Hypothesis H25 : SNo (g x x2)

L4345

Hypothesis H26 : SNo (g u v)

L4346

Hypothesis H27 : SNo (g u x2)

L4347

Hypothesis H28 : SNo (g u y)

L4348

Hypothesis H29 : SNo (g x2 z)

L4349

Hypothesis H30 : SNo (g y y2)

L4350

Hypothesis H31 : SNo (g u (g x2 z))

L4351

Hypothesis H32 : SNo (g u (g y y2))

L4352

Hypothesis H33 : SNo (g x2 y2)

L4353

Hypothesis H34 : SNo (g x (g x2 y2))

L4354

Hypothesis H35 : SNo (g x (g x2 z))

L4355

Hypothesis H36 : SNo (g x (g y y2))

L4356

Theorem. (Conj_mul_SNo_assoc_lem2__9__21)

SNo (g u (g y z) + g x v) → g (g x y) z < w

In Proofgold the corresponding term root is 5b2894... and proposition id is c607d2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__10__9

L4362

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__10__9 TMGsi1FDPE6fZuLDFSKAeURsuvUrm7ndu25 bounty of about 25 bars ***)

L4363

Variable x : set

L4364

Variable y : set

L4365

Variable z : set

L4366

Variable w : set

L4367

Variable u : set

L4368

Variable v : set

L4369

Variable x2 : set

L4370

Variable y2 : set

L4371

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4372

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4373

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4374

Hypothesis H3 : SNo x

L4375

Hypothesis H4 : SNo z

L4376

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4377

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4378

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4379

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4380

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4381

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4382

Hypothesis H12 : SNo (g x y)

L4383

Hypothesis H13 : SNo (g (g x y) z)

L4384

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4385

Hypothesis H15 : SNo v

L4386

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4387

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4388

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4389

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4390

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4391

Hypothesis H21 : SNo u

L4392

Hypothesis H22 : SNo y2

L4393

Hypothesis H23 : SNo (g u (g y z))

L4394

Hypothesis H24 : SNo (g x v)

L4395

Hypothesis H25 : SNo (g x x2)

L4396

Hypothesis H26 : SNo (g u v)

L4397

Hypothesis H27 : SNo (g u x2)

L4398

Hypothesis H28 : SNo (g u y)

L4399

Hypothesis H29 : SNo (g x2 z)

L4400

Hypothesis H30 : SNo (g y y2)

L4401

Hypothesis H31 : SNo (g u (g x2 z))

L4402

Hypothesis H32 : SNo (g u (g y y2))

L4403

Hypothesis H33 : SNo (g x2 y2)

L4404

Hypothesis H34 : SNo (g x (g x2 y2))

L4405

Hypothesis H35 : SNo (g x (g x2 z))

L4406

Theorem. (Conj_mul_SNo_assoc_lem2__10__9)

SNo (g x (g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 190c4b... and proposition id is c4b468...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__11__1

L4412

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__11__1 TMTveihSiFsnGtaqFJJmw6SZ8gQRMUU3PeZ bounty of about 25 bars ***)

L4413

Variable x : set

L4414

Variable y : set

L4415

Variable z : set

L4416

Variable w : set

L4417

Variable u : set

L4418

Variable v : set

L4419

Variable x2 : set

L4420

Variable y2 : set

L4421

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4422

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4423

Hypothesis H3 : SNo x

L4424

Hypothesis H4 : SNo z

L4425

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4426

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4427

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4428

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4429

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4430

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4431

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4432

Hypothesis H12 : SNo (g x y)

L4433

Hypothesis H13 : SNo (g (g x y) z)

L4434

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4435

Hypothesis H15 : SNo v

L4436

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4437

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4438

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4439

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4440

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4441

Hypothesis H21 : SNo u

L4442

Hypothesis H22 : SNo y2

L4443

Hypothesis H23 : SNo (g u (g y z))

L4444

Hypothesis H24 : SNo (g x v)

L4445

Hypothesis H25 : SNo (g x x2)

L4446

Hypothesis H26 : SNo (g u v)

L4447

Hypothesis H27 : SNo (g u x2)

L4448

Hypothesis H28 : SNo (g u y)

L4449

Hypothesis H29 : SNo (g x2 z)

L4450

Hypothesis H30 : SNo (g y y2)

L4451

Hypothesis H31 : SNo (g u (g x2 z))

L4452

Hypothesis H32 : SNo (g u (g y y2))

L4453

Hypothesis H33 : SNo (g x2 y2)

L4454

Hypothesis H34 : SNo (g x (g x2 y2))

L4455

Theorem. (Conj_mul_SNo_assoc_lem2__11__1)

SNo (g x (g x2 z)) → g (g x y) z < w

In Proofgold the corresponding term root is f6c2be... and proposition id is 66b1c4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__11__19

L4461

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__11__19 TMTSX1tmosHsmCsQ4v9PawTTjvH9ogiEzR2 bounty of about 25 bars ***)

L4462

Variable x : set

L4463

Variable y : set

L4464

Variable z : set

L4465

Variable w : set

L4466

Variable u : set

L4467

Variable v : set

L4468

Variable x2 : set

L4469

Variable y2 : set

L4470

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4471

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4472

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4473

Hypothesis H3 : SNo x

L4474

Hypothesis H4 : SNo z

L4475

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4476

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4477

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4478

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4479

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4480

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4481

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4482

Hypothesis H12 : SNo (g x y)

L4483

Hypothesis H13 : SNo (g (g x y) z)

L4484

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4485

Hypothesis H15 : SNo v

L4486

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4487

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4488

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4489

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4490

Hypothesis H21 : SNo u

L4491

Hypothesis H22 : SNo y2

L4492

Hypothesis H23 : SNo (g u (g y z))

L4493

Hypothesis H24 : SNo (g x v)

L4494

Hypothesis H25 : SNo (g x x2)

L4495

Hypothesis H26 : SNo (g u v)

L4496

Hypothesis H27 : SNo (g u x2)

L4497

Hypothesis H28 : SNo (g u y)

L4498

Hypothesis H29 : SNo (g x2 z)

L4499

Hypothesis H30 : SNo (g y y2)

L4500

Hypothesis H31 : SNo (g u (g x2 z))

L4501

Hypothesis H32 : SNo (g u (g y y2))

L4502

Hypothesis H33 : SNo (g x2 y2)

L4503

Hypothesis H34 : SNo (g x (g x2 y2))

L4504

Theorem. (Conj_mul_SNo_assoc_lem2__11__19)

SNo (g x (g x2 z)) → g (g x y) z < w

In Proofgold the corresponding term root is 670786... and proposition id is fbe344...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__12__31

L4510

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__12__31 TMRKA8Gfj8o5Lh9jbByM4iTetvYEcWJ3NFG bounty of about 25 bars ***)

L4511

Variable x : set

L4512

Variable y : set

L4513

Variable z : set

L4514

Variable w : set

L4515

Variable u : set

L4516

Variable v : set

L4517

Variable x2 : set

L4518

Variable y2 : set

L4519

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4520

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4521

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4522

Hypothesis H3 : SNo x

L4523

Hypothesis H4 : SNo z

L4524

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4525

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4526

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4527

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4528

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4529

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4530

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4531

Hypothesis H12 : SNo (g x y)

L4532

Hypothesis H13 : SNo (g (g x y) z)

L4533

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4534

Hypothesis H15 : SNo v

L4535

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4536

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4537

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4538

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4539

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4540

Hypothesis H21 : SNo u

L4541

Hypothesis H22 : SNo y2

L4542

Hypothesis H23 : SNo (g u (g y z))

L4543

Hypothesis H24 : SNo (g x v)

L4544

Hypothesis H25 : SNo (g x x2)

L4545

Hypothesis H26 : SNo (g u v)

L4546

Hypothesis H27 : SNo (g u x2)

L4547

Hypothesis H28 : SNo (g u y)

L4548

Hypothesis H29 : SNo (g x2 z)

L4549

Hypothesis H30 : SNo (g y y2)

L4550

Hypothesis H32 : SNo (g u (g y y2))

L4551

Hypothesis H33 : SNo (g x2 y2)

L4552

Theorem. (Conj_mul_SNo_assoc_lem2__12__31)

SNo (g x (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 3163ca... and proposition id is a3a808...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__13__8

L4558

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__13__8 TMWkPGkTiPDRxDJUSiLm1ZX5LYbZhKtDSLw bounty of about 25 bars ***)

L4559

Variable x : set

L4560

Variable y : set

L4561

Variable z : set

L4562

Variable w : set

L4563

Variable u : set

L4564

Variable v : set

L4565

Variable x2 : set

L4566

Variable y2 : set

L4567

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4568

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4569

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4570

Hypothesis H3 : SNo x

L4571

Hypothesis H4 : SNo z

L4572

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4573

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4574

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4575

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4576

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4577

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4578

Hypothesis H12 : SNo (g x y)

L4579

Hypothesis H13 : SNo (g (g x y) z)

L4580

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4581

Hypothesis H15 : SNo v

L4582

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4583

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4584

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4585

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4586

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4587

Hypothesis H21 : SNo u

L4588

Hypothesis H22 : SNo x2

L4589

Hypothesis H23 : SNo y2

L4590

Hypothesis H24 : SNo (g u (g y z))

L4591

Hypothesis H25 : SNo (g x v)

L4592

Hypothesis H26 : SNo (g x x2)

L4593

Hypothesis H27 : SNo (g u v)

L4594

Hypothesis H28 : SNo (g u x2)

L4595

Hypothesis H29 : SNo (g u y)

L4596

Hypothesis H30 : SNo (g x2 z)

L4597

Hypothesis H31 : SNo (g y y2)

L4598

Hypothesis H32 : SNo (g u (g x2 z))

L4599

Hypothesis H33 : SNo (g u (g y y2))

L4600

Theorem. (Conj_mul_SNo_assoc_lem2__13__8)

SNo (g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 84849f... and proposition id is 54ed3e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__14__5

L4606

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__14__5 TMFDfGZNgvY7GVmvqc1Zyn2QVqfjMUAdr6W bounty of about 25 bars ***)

L4607

Variable x : set

L4608

Variable y : set

L4609

Variable z : set

L4610

Variable w : set

L4611

Variable u : set

L4612

Variable v : set

L4613

Variable x2 : set

L4614

Variable y2 : set

L4615

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4616

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4617

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4618

Hypothesis H3 : SNo x

L4619

Hypothesis H4 : SNo z

L4620

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4621

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4622

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4623

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4624

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4625

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4626

Hypothesis H12 : SNo (g x y)

L4627

Hypothesis H13 : SNo (g (g x y) z)

L4628

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4629

Hypothesis H15 : SNo v

L4630

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4631

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4632

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4633

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4634

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4635

Hypothesis H21 : SNo u

L4636

Hypothesis H22 : SNo x2

L4637

Hypothesis H23 : SNo y2

L4638

Hypothesis H24 : SNo (g u (g y z))

L4639

Hypothesis H25 : SNo (g x v)

L4640

Hypothesis H26 : SNo (g x x2)

L4641

Hypothesis H27 : SNo (g u v)

L4642

Hypothesis H28 : SNo (g u x2)

L4643

Hypothesis H29 : SNo (g u y)

L4644

Hypothesis H30 : SNo (g x2 z)

L4645

Hypothesis H31 : SNo (g y y2)

L4646

Hypothesis H32 : SNo (g u (g x2 z))

L4647

Theorem. (Conj_mul_SNo_assoc_lem2__14__5)

SNo (g u (g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 3f98da... and proposition id is e86799...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__14__29

L4653

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__14__29 TMcaPoRJV7xALRquuc4fvzKSPFk6Taqkwm5 bounty of about 25 bars ***)

L4654

Variable x : set

L4655

Variable y : set

L4656

Variable z : set

L4657

Variable w : set

L4658

Variable u : set

L4659

Variable v : set

L4660

Variable x2 : set

L4661

Variable y2 : set

L4662

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4663

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4664

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4665

Hypothesis H3 : SNo x

L4666

Hypothesis H4 : SNo z

L4667

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4668

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4669

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4670

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4671

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4672

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4673

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4674

Hypothesis H12 : SNo (g x y)

L4675

Hypothesis H13 : SNo (g (g x y) z)

L4676

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4677

Hypothesis H15 : SNo v

L4678

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4679

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4680

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4681

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4682

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4683

Hypothesis H21 : SNo u

L4684

Hypothesis H22 : SNo x2

L4685

Hypothesis H23 : SNo y2

L4686

Hypothesis H24 : SNo (g u (g y z))

L4687

Hypothesis H25 : SNo (g x v)

L4688

Hypothesis H26 : SNo (g x x2)

L4689

Hypothesis H27 : SNo (g u v)

L4690

Hypothesis H28 : SNo (g u x2)

L4691

Hypothesis H30 : SNo (g x2 z)

L4692

Hypothesis H31 : SNo (g y y2)

L4693

Hypothesis H32 : SNo (g u (g x2 z))

L4694

Theorem. (Conj_mul_SNo_assoc_lem2__14__29)

SNo (g u (g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is cc3686... and proposition id is 937f0c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__15__1

L4700

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__15__1 TMEnubJbuBU5CAKZN7nK1VDXT4xzTWdoWpm bounty of about 25 bars ***)

L4701

Variable x : set

L4702

Variable y : set

L4703

Variable z : set

L4704

Variable w : set

L4705

Variable u : set

L4706

Variable v : set

L4707

Variable x2 : set

L4708

Variable y2 : set

L4709

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4710

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4711

Hypothesis H3 : SNo x

L4712

Hypothesis H4 : SNo z

L4713

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4714

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4715

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4716

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4717

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4718

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4719

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4720

Hypothesis H12 : SNo (g x y)

L4721

Hypothesis H13 : SNo (g (g x y) z)

L4722

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4723

Hypothesis H15 : SNo v

L4724

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4725

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4726

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4727

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4728

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4729

Hypothesis H21 : SNo u

L4730

Hypothesis H22 : SNo x2

L4731

Hypothesis H23 : SNo y2

L4732

Hypothesis H24 : SNo (g u (g y z))

L4733

Hypothesis H25 : SNo (g x v)

L4734

Hypothesis H26 : SNo (g x x2)

L4735

Hypothesis H27 : SNo (g u v)

L4736

Hypothesis H28 : SNo (g u x2)

L4737

Hypothesis H29 : SNo (g u y)

L4738

Hypothesis H30 : SNo (g x2 z)

L4739

Hypothesis H31 : SNo (g y y2)

L4740

Theorem. (Conj_mul_SNo_assoc_lem2__15__1)

SNo (g u (g x2 z)) → g (g x y) z < w

In Proofgold the corresponding term root is 4ec8e2... and proposition id is 2e25ae...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__15__27

L4746

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__15__27 TMRjvQf8C9ihFkoM9LQY2abJ51zKaggcRJV bounty of about 25 bars ***)

L4747

Variable x : set

L4748

Variable y : set

L4749

Variable z : set

L4750

Variable w : set

L4751

Variable u : set

L4752

Variable v : set

L4753

Variable x2 : set

L4754

Variable y2 : set

L4755

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4756

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4757

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4758

Hypothesis H3 : SNo x

L4759

Hypothesis H4 : SNo z

L4760

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4761

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4762

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4763

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4764

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4765

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4766

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4767

Hypothesis H12 : SNo (g x y)

L4768

Hypothesis H13 : SNo (g (g x y) z)

L4769

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4770

Hypothesis H15 : SNo v

L4771

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4772

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4773

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4774

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4775

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4776

Hypothesis H21 : SNo u

L4777

Hypothesis H22 : SNo x2

L4778

Hypothesis H23 : SNo y2

L4779

Hypothesis H24 : SNo (g u (g y z))

L4780

Hypothesis H25 : SNo (g x v)

L4781

Hypothesis H26 : SNo (g x x2)

L4782

Hypothesis H28 : SNo (g u x2)

L4783

Hypothesis H29 : SNo (g u y)

L4784

Hypothesis H30 : SNo (g x2 z)

L4785

Hypothesis H31 : SNo (g y y2)

L4786

Theorem. (Conj_mul_SNo_assoc_lem2__15__27)

SNo (g u (g x2 z)) → g (g x y) z < w

In Proofgold the corresponding term root is be03be... and proposition id is ccf2d9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__15__29

L4792

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__15__29 TMTB4dxGBwZyxW5iqnwtzL6MhKPx3NmtZPJ bounty of about 25 bars ***)

L4793

Variable x : set

L4794

Variable y : set

L4795

Variable z : set

L4796

Variable w : set

L4797

Variable u : set

L4798

Variable v : set

L4799

Variable x2 : set

L4800

Variable y2 : set

L4801

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4802

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4803

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4804

Hypothesis H3 : SNo x

L4805

Hypothesis H4 : SNo z

L4806

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4807

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4808

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4809

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4810

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4811

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4812

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4813

Hypothesis H12 : SNo (g x y)

L4814

Hypothesis H13 : SNo (g (g x y) z)

L4815

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L4816

Hypothesis H15 : SNo v

L4817

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L4818

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L4819

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L4820

Hypothesis H19 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4821

Hypothesis H20 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4822

Hypothesis H21 : SNo u

L4823

Hypothesis H22 : SNo x2

L4824

Hypothesis H23 : SNo y2

L4825

Hypothesis H24 : SNo (g u (g y z))

L4826

Hypothesis H25 : SNo (g x v)

L4827

Hypothesis H26 : SNo (g x x2)

L4828

Hypothesis H27 : SNo (g u v)

L4829

Hypothesis H28 : SNo (g u x2)

L4830

Hypothesis H30 : SNo (g x2 z)

L4831

Hypothesis H31 : SNo (g y y2)

L4832

Theorem. (Conj_mul_SNo_assoc_lem2__15__29)

SNo (g u (g x2 z)) → g (g x y) z < w

In Proofgold the corresponding term root is a6f9c9... and proposition id is 796304...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__18__13

L4838

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__18__13 TMSbJEdibeqbZk8B7cPRrvMibL5A1in8nnp bounty of about 25 bars ***)

L4839

Variable x : set

L4840

Variable y : set

L4841

Variable z : set

L4842

Variable w : set

L4843

Variable u : set

L4844

Variable v : set

L4845

Variable x2 : set

L4846

Variable y2 : set

L4847

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4848

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4849

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4850

Hypothesis H3 : SNo x

L4851

Hypothesis H4 : SNo y

L4852

Hypothesis H5 : SNo z

L4853

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4854

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4855

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4856

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4857

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4858

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4859

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4860

Hypothesis H14 : SNo (g (g x y) z)

L4861

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L4862

Hypothesis H16 : SNo v

L4863

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L4864

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L4865

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L4866

Hypothesis H20 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4867

Hypothesis H21 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4868

Hypothesis H22 : SNo u

L4869

Hypothesis H23 : SNo x2

L4870

Hypothesis H24 : SNo y2

L4871

Hypothesis H25 : SNo (g u (g y z))

L4872

Hypothesis H26 : SNo (g x v)

L4873

Hypothesis H27 : SNo (g x x2)

L4874

Hypothesis H28 : SNo (g u v)

L4875

Hypothesis H29 : SNo (g u x2)

L4876

Theorem. (Conj_mul_SNo_assoc_lem2__18__13)

SNo (g u y) → g (g x y) z < w

In Proofgold the corresponding term root is 9e1ad7... and proposition id is 5a8407...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__18__18

L4882

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__18__18 TMRhSfNXEsoJZUM4C3Z73uop8S9LmHBqBim bounty of about 25 bars ***)

L4883

Variable x : set

L4884

Variable y : set

L4885

Variable z : set

L4886

Variable w : set

L4887

Variable u : set

L4888

Variable v : set

L4889

Variable x2 : set

L4890

Variable y2 : set

L4891

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4892

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4893

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4894

Hypothesis H3 : SNo x

L4895

Hypothesis H4 : SNo y

L4896

Hypothesis H5 : SNo z

L4897

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4898

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4899

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4900

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4901

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4902

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4903

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4904

Hypothesis H13 : SNo (g x y)

L4905

Hypothesis H14 : SNo (g (g x y) z)

L4906

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L4907

Hypothesis H16 : SNo v

L4908

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L4909

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L4910

Hypothesis H20 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4911

Hypothesis H21 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4912

Hypothesis H22 : SNo u

L4913

Hypothesis H23 : SNo x2

L4914

Hypothesis H24 : SNo y2

L4915

Hypothesis H25 : SNo (g u (g y z))

L4916

Hypothesis H26 : SNo (g x v)

L4917

Hypothesis H27 : SNo (g x x2)

L4918

Hypothesis H28 : SNo (g u v)

L4919

Hypothesis H29 : SNo (g u x2)

L4920

Theorem. (Conj_mul_SNo_assoc_lem2__18__18)

SNo (g u y) → g (g x y) z < w

In Proofgold the corresponding term root is 9e6d15... and proposition id is e8c522...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__19__8

L4926

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__19__8 TMPAiERC5n5Hf5zKrUYxbkJYCEKRk9qzXcG bounty of about 25 bars ***)

L4927

Variable x : set

L4928

Variable y : set

L4929

Variable z : set

L4930

Variable w : set

L4931

Variable u : set

L4932

Variable v : set

L4933

Variable x2 : set

L4934

Variable y2 : set

L4935

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4936

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4937

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4938

Hypothesis H3 : SNo x

L4939

Hypothesis H4 : SNo y

L4940

Hypothesis H5 : SNo z

L4941

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4942

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4943

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4944

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4945

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L4946

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4947

Hypothesis H13 : SNo (g x y)

L4948

Hypothesis H14 : SNo (g (g x y) z)

L4949

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L4950

Hypothesis H16 : SNo v

L4951

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L4952

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L4953

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L4954

Hypothesis H20 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4955

Hypothesis H21 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4956

Hypothesis H22 : SNo u

L4957

Hypothesis H23 : SNo x2

L4958

Hypothesis H24 : SNo y2

L4959

Hypothesis H25 : SNo (g u (g y z))

L4960

Hypothesis H26 : SNo (g x v)

L4961

Hypothesis H27 : SNo (g x x2)

L4962

Hypothesis H28 : SNo (g u v)

L4963

Theorem. (Conj_mul_SNo_assoc_lem2__19__8)

SNo (g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is 88f55a... and proposition id is 0b1541...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__19__11

L4969

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__19__11 TMdPtmSG5J2dYTm8Dc17zeXGj9vMgRUDCpY bounty of about 25 bars ***)

L4970

Variable x : set

L4971

Variable y : set

L4972

Variable z : set

L4973

Variable w : set

L4974

Variable u : set

L4975

Variable v : set

L4976

Variable x2 : set

L4977

Variable y2 : set

L4978

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L4979

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L4980

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L4981

Hypothesis H3 : SNo x

L4982

Hypothesis H4 : SNo y

L4983

Hypothesis H5 : SNo z

L4984

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L4985

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L4986

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L4987

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L4988

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L4989

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L4990

Hypothesis H13 : SNo (g x y)

L4991

Hypothesis H14 : SNo (g (g x y) z)

L4992

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L4993

Hypothesis H16 : SNo v

L4994

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L4995

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L4996

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L4997

Hypothesis H20 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L4998

Hypothesis H21 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L4999

Hypothesis H22 : SNo u

L5000

Hypothesis H23 : SNo x2

L5001

Hypothesis H24 : SNo y2

L5002

Hypothesis H25 : SNo (g u (g y z))

L5003

Hypothesis H26 : SNo (g x v)

L5004

Hypothesis H27 : SNo (g x x2)

L5005

Hypothesis H28 : SNo (g u v)

L5006

Theorem. (Conj_mul_SNo_assoc_lem2__19__11)

SNo (g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is bd4222... and proposition id is 8ab45d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__21__8

L5012

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__21__8 TMSdGUZPXx5KF7bD7satVPWQhyKc7gxQFNq bounty of about 25 bars ***)

L5013

Variable x : set

L5014

Variable y : set

L5015

Variable z : set

L5016

Variable w : set

L5017

Variable u : set

L5018

Variable v : set

L5019

Variable x2 : set

L5020

Variable y2 : set

L5021

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5022

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L5023

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L5024

Hypothesis H3 : SNo x

L5025

Hypothesis H4 : SNo y

L5026

Hypothesis H5 : SNo z

L5027

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L5028

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L5029

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L5030

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L5031

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L5032

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L5033

Hypothesis H13 : SNo (g x y)

L5034

Hypothesis H14 : SNo (g (g x y) z)

L5035

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L5036

Hypothesis H16 : SNo v

L5037

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L5038

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L5039

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L5040

Hypothesis H20 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L5041

Hypothesis H21 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L5042

Hypothesis H22 : SNo u

L5043

Hypothesis H23 : SNo x2

L5044

Hypothesis H24 : SNo y2

L5045

Hypothesis H25 : SNo (g u (g y z))

L5046

Hypothesis H26 : SNo (g x v)

L5047

Theorem. (Conj_mul_SNo_assoc_lem2__21__8)

SNo (g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is e0f87d... and proposition id is 4a1d4d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__22__22

L5053

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__22__22 TMPLALSpJNjQHmDL4NFiTP7R21jpZzEMRX1 bounty of about 25 bars ***)

L5054

Variable x : set

L5055

Variable y : set

L5056

Variable z : set

L5057

Variable w : set

L5058

Variable u : set

L5059

Variable v : set

L5060

Variable x2 : set

L5061

Variable y2 : set

L5062

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5063

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L5064

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L5065

Hypothesis H3 : SNo x

L5066

Hypothesis H4 : SNo y

L5067

Hypothesis H5 : SNo z

L5068

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L5069

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L5070

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L5071

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L5072

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L5073

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L5074

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L5075

Hypothesis H13 : SNo (g x y)

L5076

Hypothesis H14 : SNo (g (g x y) z)

L5077

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L5078

Hypothesis H16 : SNo v

L5079

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L5080

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L5081

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L5082

Hypothesis H20 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L5083

Hypothesis H21 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L5084

Hypothesis H23 : SNo x2

L5085

Hypothesis H24 : SNo y2

L5086

Hypothesis H25 : SNo (g u (g y z))

L5087

Theorem. (Conj_mul_SNo_assoc_lem2__22__22)

SNo (g x v) → g (g x y) z < w

In Proofgold the corresponding term root is ce5a88... and proposition id is 8207ce...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__23__13

L5093

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__23__13 TMGzC8yB5kEaE8wYXh2nFEKULRt6zJmER5i bounty of about 25 bars ***)

L5094

Variable x : set

L5095

Variable y : set

L5096

Variable z : set

L5097

Variable w : set

L5098

Variable u : set

L5099

Variable v : set

L5100

Variable x2 : set

L5101

Variable y2 : set

L5102

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5103

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L5104

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L5105

Hypothesis H3 : SNo x

L5106

Hypothesis H4 : SNo y

L5107

Hypothesis H5 : SNo z

L5108

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L5109

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L5110

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L5111

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L5112

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L5113

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L5114

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L5115

Hypothesis H14 : SNo (g y z)

L5116

Hypothesis H15 : SNo (g (g x y) z)

L5117

Hypothesis H16 : u ∈ SNoS_ (SNoLev x)

L5118

Hypothesis H17 : SNo v

L5119

Hypothesis H18 : x2 ∈ SNoS_ (SNoLev y)

L5120

Hypothesis H19 : y2 ∈ SNoS_ (SNoLev z)

L5121

Hypothesis H20 : w = g u (g y z) + g x v + - (g u v)

L5122

Hypothesis H21 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L5123

Hypothesis H22 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L5124

Hypothesis H23 : SNo u

L5125

Hypothesis H24 : SNo x2

L5126

Hypothesis H25 : SNo y2

L5127

Theorem. (Conj_mul_SNo_assoc_lem2__23__13)

SNo (g u (g y z)) → g (g x y) z < w

In Proofgold the corresponding term root is 9410b7... and proposition id is 8f33b3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__23__23

L5133

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__23__23 TMZS7pvNKGLAzG1DCi5HJpjPY9xMgRTndSF bounty of about 25 bars ***)

L5134

Variable x : set

L5135

Variable y : set

L5136

Variable z : set

L5137

Variable w : set

L5138

Variable u : set

L5139

Variable v : set

L5140

Variable x2 : set

L5141

Variable y2 : set

L5142

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5143

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L5144

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g (z2 + w2) u2 = g z2 u2 + g w2 u2)

L5145

Hypothesis H3 : SNo x

L5146

Hypothesis H4 : SNo y

L5147

Hypothesis H5 : SNo z

L5148

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L5149

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L5150

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L5151

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L5152

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L5153

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L5154

Hypothesis H12 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → (∀u2 : set, u2 ∈ SNoS_ (SNoLev z) → g z2 (g w2 u2) = g (g z2 w2) u2)))

L5155

Hypothesis H13 : SNo (g x y)

L5156

Hypothesis H14 : SNo (g y z)

L5157

Hypothesis H15 : SNo (g (g x y) z)

L5158

Hypothesis H16 : u ∈ SNoS_ (SNoLev x)

L5159

Hypothesis H17 : SNo v

L5160

Hypothesis H18 : x2 ∈ SNoS_ (SNoLev y)

L5161

Hypothesis H19 : y2 ∈ SNoS_ (SNoLev z)

L5162

Hypothesis H20 : w = g u (g y z) + g x v + - (g u v)

L5163

Hypothesis H21 : (g x (g x2 z + g y y2) + g u (v + g x2 y2)) ≤ g u (g x2 z + g y y2) + g x (v + g x2 y2)

L5164

Hypothesis H22 : (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2

L5165

Hypothesis H24 : SNo x2

L5166

Hypothesis H25 : SNo y2

L5167

Theorem. (Conj_mul_SNo_assoc_lem2__23__23)

SNo (g u (g y z)) → g (g x y) z < w

In Proofgold the corresponding term root is 310b90... and proposition id is ca9b1c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__24__1

L5173

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__24__1 TMN1TZQea8oMprwHKLLtA8Fj69vA3WjDGCD bounty of about 25 bars ***)

L5174

Variable x : set

L5175

Variable y : set

L5176

Variable z : set

L5177

Variable w : set

L5178

Variable u : set

L5179

Variable v : set

L5180

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L5181

Hypothesis H2 : SNo x

L5182

Hypothesis H3 : SNo y

L5183

Hypothesis H4 : SNo z

L5184

Hypothesis H5 : SNo (g x y)

L5185

Hypothesis H6 : SNo w

L5186

Hypothesis H7 : x < w

L5187

Hypothesis H8 : SNo u

L5188

Hypothesis H9 : y < u

L5189

Hypothesis H10 : SNo v

L5190

Hypothesis H11 : z < v

L5191

Hypothesis H12 : SNo (g w y)

L5192

Hypothesis H13 : SNo (g w u)

L5193

Hypothesis H14 : SNo (g x u)

L5194

Hypothesis H15 : SNo (g (g w y + g x u) z)

L5195

Hypothesis H16 : SNo (g (g x y + g w u) z)

L5196

Hypothesis H17 : SNo (g (g w y + g x u) v)

L5197

Hypothesis H18 : SNo (g (g x y + g w u) v)

L5198

Hypothesis H19 : SNo (g w y + g x u)

L5199

Theorem. (Conj_mul_SNo_assoc_lem2__24__1)

SNo (g x y + g w u) → (g (g x y + g w u) z + g (g w y + g x u) v) < g (g w y + g x u) z + g (g x y + g w u) v

In Proofgold the corresponding term root is 32b0be... and proposition id is e69f53...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__25__14

L5205

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__25__14 TMQYM7NwQtJHgtiFTXYDjtAHnRkN6VEDxSW bounty of about 25 bars ***)

L5206

Variable x : set

L5207

Variable y : set

L5208

Variable z : set

L5209

Variable w : set

L5210

Variable u : set

L5211

Variable v : set

L5212

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L5213

Hypothesis H1 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L5214

Hypothesis H2 : SNo x

L5215

Hypothesis H3 : SNo y

L5216

Hypothesis H4 : SNo z

L5217

Hypothesis H5 : SNo (g x y)

L5218

Hypothesis H6 : SNo w

L5219

Hypothesis H7 : x < w

L5220

Hypothesis H8 : SNo u

L5221

Hypothesis H9 : y < u

L5222

Hypothesis H10 : SNo v

L5223

Hypothesis H11 : z < v

L5224

Hypothesis H12 : SNo (g w y)

L5225

Hypothesis H13 : SNo (g w u)

L5226

Hypothesis H15 : SNo (g (g w y + g x u) z)

L5227

Hypothesis H16 : SNo (g (g x y + g w u) z)

L5228

Hypothesis H17 : SNo (g (g w y + g x u) v)

L5229

Hypothesis H18 : SNo (g (g x y + g w u) v)

L5230

Theorem. (Conj_mul_SNo_assoc_lem2__25__14)

SNo (g w y + g x u) → (g (g x y + g w u) z + g (g w y + g x u) v) < g (g w y + g x u) z + g (g x y + g w u) v

In Proofgold the corresponding term root is 849ad2... and proposition id is f49594...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__28__17

L5236

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__28__17 TMS7QnHzYwthVFEwwZ2oHnBaxVAVysPQ5XW bounty of about 25 bars ***)

L5237

Variable x : set

L5238

Variable y : set

L5239

Variable z : set

L5240

Variable w : set

L5241

Variable u : set

L5242

Variable v : set

L5243

Variable x2 : set

L5244

Variable y2 : set

L5245

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5246

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5247

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5248

Hypothesis H3 : SNo x

L5249

Hypothesis H4 : SNo y

L5250

Hypothesis H5 : SNo z

L5251

Hypothesis H6 : SNo (g x y)

L5252

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5253

Hypothesis H8 : u ∈ SNoR x

L5254

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5255

Hypothesis H10 : SNo u

L5256

Hypothesis H11 : x < u

L5257

Hypothesis H12 : SNo v

L5258

Hypothesis H13 : x2 ∈ SNoR y

L5259

Hypothesis H14 : y2 ∈ SNoR z

L5260

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L5261

Hypothesis H16 : SNo x2

L5262

Hypothesis H18 : SNo y2

L5263

Hypothesis H19 : z < y2

L5264

Hypothesis H20 : SNo (g u y)

L5265

Hypothesis H21 : SNo (g u x2)

L5266

Hypothesis H22 : SNo (g x x2)

L5267

Hypothesis H23 : SNo (g x2 z + g y y2)

L5268

Hypothesis H24 : SNo (v + g x2 y2)

L5269

Hypothesis H25 : SNo (g u y + g x x2)

L5270

Hypothesis H26 : SNo (g x y + g u x2)

L5271

Hypothesis H27 : SNo (g (g u y + g x x2) z)

L5272

Theorem. (Conj_mul_SNo_assoc_lem2__28__17)

SNo (g (g x y + g u x2) z) → g (g x y) z < w

In Proofgold the corresponding term root is 559b7e... and proposition id is d4174d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__31__23

L5278

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__31__23 TMaswt3VQfgAP8iLwNQf3SddDJMfXKuq9jE bounty of about 25 bars ***)

L5279

Variable x : set

L5280

Variable y : set

L5281

Variable z : set

L5282

Variable w : set

L5283

Variable u : set

L5284

Variable v : set

L5285

Variable x2 : set

L5286

Variable y2 : set

L5287

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5288

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5289

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5290

Hypothesis H3 : SNo x

L5291

Hypothesis H4 : SNo y

L5292

Hypothesis H5 : SNo z

L5293

Hypothesis H6 : SNo (g x y)

L5294

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5295

Hypothesis H8 : u ∈ SNoR x

L5296

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5297

Hypothesis H10 : SNo u

L5298

Hypothesis H11 : x < u

L5299

Hypothesis H12 : SNo v

L5300

Hypothesis H13 : x2 ∈ SNoR y

L5301

Hypothesis H14 : y2 ∈ SNoR z

L5302

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L5303

Hypothesis H16 : SNo x2

L5304

Hypothesis H17 : y < x2

L5305

Hypothesis H18 : SNo y2

L5306

Hypothesis H19 : z < y2

L5307

Hypothesis H20 : SNo (g u y)

L5308

Hypothesis H21 : SNo (g u x2)

L5309

Hypothesis H22 : SNo (g x x2)

L5310

Hypothesis H24 : SNo (v + g x2 y2)

L5311

Theorem. (Conj_mul_SNo_assoc_lem2__31__23)

SNo (g u y + g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 573324... and proposition id is a10eb2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__32__4

L5317

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__32__4 TMTUL9hvCGi16XsBkqVAzMAGyLmHNXjcHfk bounty of about 25 bars ***)

L5318

Variable x : set

L5319

Variable y : set

L5320

Variable z : set

L5321

Variable w : set

L5322

Variable u : set

L5323

Variable v : set

L5324

Variable x2 : set

L5325

Variable y2 : set

L5326

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5327

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5328

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5329

Hypothesis H3 : SNo x

L5330

Hypothesis H5 : SNo z

L5331

Hypothesis H6 : SNo (g x y)

L5332

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5333

Hypothesis H8 : u ∈ SNoR x

L5334

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5335

Hypothesis H10 : SNo u

L5336

Hypothesis H11 : x < u

L5337

Hypothesis H12 : SNo v

L5338

Hypothesis H13 : x2 ∈ SNoR y

L5339

Hypothesis H14 : y2 ∈ SNoR z

L5340

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L5341

Hypothesis H16 : SNo x2

L5342

Hypothesis H17 : y < x2

L5343

Hypothesis H18 : SNo y2

L5344

Hypothesis H19 : z < y2

L5345

Hypothesis H20 : SNo (g u y)

L5346

Hypothesis H21 : SNo (g u x2)

L5347

Hypothesis H22 : SNo (g x x2)

L5348

Hypothesis H23 : SNo (g x2 z + g y y2)

L5349

Hypothesis H24 : SNo (v + g x2 y2)

L5350

Theorem. (Conj_mul_SNo_assoc_lem2__32__4)

SNo (g x (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 465215... and proposition id is 5ef8d8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__32__6

L5356

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__32__6 TMVUeBHCw5rfBG6Nf4qV3BQUp9gWJnd6KwA bounty of about 25 bars ***)

L5357

Variable x : set

L5358

Variable y : set

L5359

Variable z : set

L5360

Variable w : set

L5361

Variable u : set

L5362

Variable v : set

L5363

Variable x2 : set

L5364

Variable y2 : set

L5365

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5366

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5367

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5368

Hypothesis H3 : SNo x

L5369

Hypothesis H4 : SNo y

L5370

Hypothesis H5 : SNo z

L5371

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5372

Hypothesis H8 : u ∈ SNoR x

L5373

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5374

Hypothesis H10 : SNo u

L5375

Hypothesis H11 : x < u

L5376

Hypothesis H12 : SNo v

L5377

Hypothesis H13 : x2 ∈ SNoR y

L5378

Hypothesis H14 : y2 ∈ SNoR z

L5379

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L5380

Hypothesis H16 : SNo x2

L5381

Hypothesis H17 : y < x2

L5382

Hypothesis H18 : SNo y2

L5383

Hypothesis H19 : z < y2

L5384

Hypothesis H20 : SNo (g u y)

L5385

Hypothesis H21 : SNo (g u x2)

L5386

Hypothesis H22 : SNo (g x x2)

L5387

Hypothesis H23 : SNo (g x2 z + g y y2)

L5388

Hypothesis H24 : SNo (v + g x2 y2)

L5389

Theorem. (Conj_mul_SNo_assoc_lem2__32__6)

SNo (g x (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 795d8c... and proposition id is 9b2536...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__32__11

L5395

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__32__11 TMYfwfTdbFtNh5e1TEeEXUSYfFCjAs72MGW bounty of about 25 bars ***)

L5396

Variable x : set

L5397

Variable y : set

L5398

Variable z : set

L5399

Variable w : set

L5400

Variable u : set

L5401

Variable v : set

L5402

Variable x2 : set

L5403

Variable y2 : set

L5404

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5405

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5406

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5407

Hypothesis H3 : SNo x

L5408

Hypothesis H4 : SNo y

L5409

Hypothesis H5 : SNo z

L5410

Hypothesis H6 : SNo (g x y)

L5411

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5412

Hypothesis H8 : u ∈ SNoR x

L5413

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5414

Hypothesis H10 : SNo u

L5415

Hypothesis H12 : SNo v

L5416

Hypothesis H13 : x2 ∈ SNoR y

L5417

Hypothesis H14 : y2 ∈ SNoR z

L5418

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L5419

Hypothesis H16 : SNo x2

L5420

Hypothesis H17 : y < x2

L5421

Hypothesis H18 : SNo y2

L5422

Hypothesis H19 : z < y2

L5423

Hypothesis H20 : SNo (g u y)

L5424

Hypothesis H21 : SNo (g u x2)

L5425

Hypothesis H22 : SNo (g x x2)

L5426

Hypothesis H23 : SNo (g x2 z + g y y2)

L5427

Hypothesis H24 : SNo (v + g x2 y2)

L5428

Theorem. (Conj_mul_SNo_assoc_lem2__32__11)

SNo (g x (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 22c013... and proposition id is 9e652e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__32__21

L5434

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__32__21 TMK6mYMC2St2bcXgkUfHaZuPW38AXvVigcQ bounty of about 25 bars ***)

L5435

Variable x : set

L5436

Variable y : set

L5437

Variable z : set

L5438

Variable w : set

L5439

Variable u : set

L5440

Variable v : set

L5441

Variable x2 : set

L5442

Variable y2 : set

L5443

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5444

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5445

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5446

Hypothesis H3 : SNo x

L5447

Hypothesis H4 : SNo y

L5448

Hypothesis H5 : SNo z

L5449

Hypothesis H6 : SNo (g x y)

L5450

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5451

Hypothesis H8 : u ∈ SNoR x

L5452

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5453

Hypothesis H10 : SNo u

L5454

Hypothesis H11 : x < u

L5455

Hypothesis H12 : SNo v

L5456

Hypothesis H13 : x2 ∈ SNoR y

L5457

Hypothesis H14 : y2 ∈ SNoR z

L5458

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L5459

Hypothesis H16 : SNo x2

L5460

Hypothesis H17 : y < x2

L5461

Hypothesis H18 : SNo y2

L5462

Hypothesis H19 : z < y2

L5463

Hypothesis H20 : SNo (g u y)

L5464

Hypothesis H22 : SNo (g x x2)

L5465

Hypothesis H23 : SNo (g x2 z + g y y2)

L5466

Hypothesis H24 : SNo (v + g x2 y2)

L5467

Theorem. (Conj_mul_SNo_assoc_lem2__32__21)

SNo (g x (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is c2bf88... and proposition id is afeffc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__33__1

L5473

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__33__1 TMcCgYG3nv7wt2sXCfKPqGbyB1XZDRRQw6Z bounty of about 25 bars ***)

L5474

Variable x : set

L5475

Variable y : set

L5476

Variable z : set

L5477

Variable w : set

L5478

Variable u : set

L5479

Variable v : set

L5480

Variable x2 : set

L5481

Variable y2 : set

L5482

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5483

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5484

Hypothesis H3 : SNo x

L5485

Hypothesis H4 : SNo y

L5486

Hypothesis H5 : SNo z

L5487

Hypothesis H6 : SNo (g x y)

L5488

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5489

Hypothesis H8 : u ∈ SNoR x

L5490

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5491

Hypothesis H10 : SNo u

L5492

Hypothesis H11 : x < u

L5493

Hypothesis H12 : SNo v

L5494

Hypothesis H13 : SNo (g u v)

L5495

Hypothesis H14 : x2 ∈ SNoR y

L5496

Hypothesis H15 : y2 ∈ SNoR z

L5497

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5498

Hypothesis H17 : SNo x2

L5499

Hypothesis H18 : y < x2

L5500

Hypothesis H19 : SNo y2

L5501

Hypothesis H20 : z < y2

L5502

Hypothesis H21 : SNo (g u y)

L5503

Hypothesis H22 : SNo (g u x2)

L5504

Hypothesis H23 : SNo (g x x2)

L5505

Hypothesis H24 : SNo (g x2 z + g y y2)

L5506

Hypothesis H25 : SNo (v + g x2 y2)

L5507

Hypothesis H26 : SNo (g u (g x2 y2))

L5508

Theorem. (Conj_mul_SNo_assoc_lem2__33__1)

SNo (g u v + g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 517e3d... and proposition id is 6fa8cf...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__34__7

L5514

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__34__7 TMM2rvruYoGsNTRc2KH1P2nHUFBL5Mda1ZZ bounty of about 25 bars ***)

L5515

Variable x : set

L5516

Variable y : set

L5517

Variable z : set

L5518

Variable w : set

L5519

Variable u : set

L5520

Variable v : set

L5521

Variable x2 : set

L5522

Variable y2 : set

L5523

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5524

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5525

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5526

Hypothesis H3 : SNo x

L5527

Hypothesis H4 : SNo y

L5528

Hypothesis H5 : SNo z

L5529

Hypothesis H6 : SNo (g x y)

L5530

Hypothesis H8 : u ∈ SNoR x

L5531

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5532

Hypothesis H10 : SNo u

L5533

Hypothesis H11 : x < u

L5534

Hypothesis H12 : SNo v

L5535

Hypothesis H13 : SNo (g u v)

L5536

Hypothesis H14 : x2 ∈ SNoR y

L5537

Hypothesis H15 : y2 ∈ SNoR z

L5538

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5539

Hypothesis H17 : SNo x2

L5540

Hypothesis H18 : y < x2

L5541

Hypothesis H19 : SNo y2

L5542

Hypothesis H20 : z < y2

L5543

Hypothesis H21 : SNo (g u y)

L5544

Hypothesis H22 : SNo (g u x2)

L5545

Hypothesis H23 : SNo (g x x2)

L5546

Hypothesis H24 : SNo (g x2 z + g y y2)

L5547

Hypothesis H25 : SNo (v + g x2 y2)

L5548

Hypothesis H26 : SNo (g u (g x2 y2))

L5549

Theorem. (Conj_mul_SNo_assoc_lem2__34__7)

SNo (g u (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 67c49c... and proposition id is 7fe4af...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__36__5

L5555

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__36__5 TMKNXY6wqnYGPwAVaELvA3rrUhCLKyu2EFg bounty of about 25 bars ***)

L5556

Variable x : set

L5557

Variable y : set

L5558

Variable z : set

L5559

Variable w : set

L5560

Variable u : set

L5561

Variable v : set

L5562

Variable x2 : set

L5563

Variable y2 : set

L5564

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5565

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5566

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5567

Hypothesis H3 : SNo x

L5568

Hypothesis H4 : SNo y

L5569

Hypothesis H6 : SNo (g x y)

L5570

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5571

Hypothesis H8 : u ∈ SNoR x

L5572

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5573

Hypothesis H10 : SNo u

L5574

Hypothesis H11 : x < u

L5575

Hypothesis H12 : SNo v

L5576

Hypothesis H13 : SNo (g u v)

L5577

Hypothesis H14 : x2 ∈ SNoR y

L5578

Hypothesis H15 : y2 ∈ SNoR z

L5579

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5580

Hypothesis H17 : SNo x2

L5581

Hypothesis H18 : y < x2

L5582

Hypothesis H19 : SNo y2

L5583

Hypothesis H20 : z < y2

L5584

Hypothesis H21 : SNo (g u y)

L5585

Hypothesis H22 : SNo (g u x2)

L5586

Hypothesis H23 : SNo (g x x2)

L5587

Hypothesis H24 : SNo (g x2 z + g y y2)

L5588

Hypothesis H25 : SNo (g x2 y2)

L5589

Hypothesis H26 : SNo (v + g x2 y2)

L5590

Theorem. (Conj_mul_SNo_assoc_lem2__36__5)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 369f8c... and proposition id is a9b9c1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__36__10

L5596

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__36__10 TMYjpptZ1Kj2oLphKLafKJpYBXBfP9rBPcx bounty of about 25 bars ***)

L5597

Variable x : set

L5598

Variable y : set

L5599

Variable z : set

L5600

Variable w : set

L5601

Variable u : set

L5602

Variable v : set

L5603

Variable x2 : set

L5604

Variable y2 : set

L5605

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5606

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5607

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5608

Hypothesis H3 : SNo x

L5609

Hypothesis H4 : SNo y

L5610

Hypothesis H5 : SNo z

L5611

Hypothesis H6 : SNo (g x y)

L5612

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5613

Hypothesis H8 : u ∈ SNoR x

L5614

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5615

Hypothesis H11 : x < u

L5616

Hypothesis H12 : SNo v

L5617

Hypothesis H13 : SNo (g u v)

L5618

Hypothesis H14 : x2 ∈ SNoR y

L5619

Hypothesis H15 : y2 ∈ SNoR z

L5620

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5621

Hypothesis H17 : SNo x2

L5622

Hypothesis H18 : y < x2

L5623

Hypothesis H19 : SNo y2

L5624

Hypothesis H20 : z < y2

L5625

Hypothesis H21 : SNo (g u y)

L5626

Hypothesis H22 : SNo (g u x2)

L5627

Hypothesis H23 : SNo (g x x2)

L5628

Hypothesis H24 : SNo (g x2 z + g y y2)

L5629

Hypothesis H25 : SNo (g x2 y2)

L5630

Hypothesis H26 : SNo (v + g x2 y2)

L5631

Theorem. (Conj_mul_SNo_assoc_lem2__36__10)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 88bb6b... and proposition id is fd3bcc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__37__9

L5637

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__37__9 TMdeLPLPwDrFwU3RbWN3jfBwHswZCpjEQDZ bounty of about 25 bars ***)

L5638

Variable x : set

L5639

Variable y : set

L5640

Variable z : set

L5641

Variable w : set

L5642

Variable u : set

L5643

Variable v : set

L5644

Variable x2 : set

L5645

Variable y2 : set

L5646

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5647

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5648

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5649

Hypothesis H3 : SNo x

L5650

Hypothesis H4 : SNo y

L5651

Hypothesis H5 : SNo z

L5652

Hypothesis H6 : SNo (g x y)

L5653

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5654

Hypothesis H8 : u ∈ SNoR x

L5655

Hypothesis H10 : SNo u

L5656

Hypothesis H11 : x < u

L5657

Hypothesis H12 : SNo v

L5658

Hypothesis H13 : SNo (g u v)

L5659

Hypothesis H14 : x2 ∈ SNoR y

L5660

Hypothesis H15 : y2 ∈ SNoR z

L5661

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5662

Hypothesis H17 : SNo x2

L5663

Hypothesis H18 : y < x2

L5664

Hypothesis H19 : SNo y2

L5665

Hypothesis H20 : z < y2

L5666

Hypothesis H21 : SNo (g u y)

L5667

Hypothesis H22 : SNo (g u x2)

L5668

Hypothesis H23 : SNo (g x x2)

L5669

Hypothesis H24 : SNo (g x2 z + g y y2)

L5670

Hypothesis H25 : SNo (g x2 y2)

L5671

Hypothesis H26 : SNo (v + g x2 y2)

L5672

Theorem. (Conj_mul_SNo_assoc_lem2__37__9)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 5833de... and proposition id is 83ef7a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__37__12

L5678

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__37__12 TMNryvsbv5fFpSkCs1auQHeYpVHWWHpHhdC bounty of about 25 bars ***)

L5679

Variable x : set

L5680

Variable y : set

L5681

Variable z : set

L5682

Variable w : set

L5683

Variable u : set

L5684

Variable v : set

L5685

Variable x2 : set

L5686

Variable y2 : set

L5687

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5688

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5689

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5690

Hypothesis H3 : SNo x

L5691

Hypothesis H4 : SNo y

L5692

Hypothesis H5 : SNo z

L5693

Hypothesis H6 : SNo (g x y)

L5694

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5695

Hypothesis H8 : u ∈ SNoR x

L5696

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5697

Hypothesis H10 : SNo u

L5698

Hypothesis H11 : x < u

L5699

Hypothesis H13 : SNo (g u v)

L5700

Hypothesis H14 : x2 ∈ SNoR y

L5701

Hypothesis H15 : y2 ∈ SNoR z

L5702

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5703

Hypothesis H17 : SNo x2

L5704

Hypothesis H18 : y < x2

L5705

Hypothesis H19 : SNo y2

L5706

Hypothesis H20 : z < y2

L5707

Hypothesis H21 : SNo (g u y)

L5708

Hypothesis H22 : SNo (g u x2)

L5709

Hypothesis H23 : SNo (g x x2)

L5710

Hypothesis H24 : SNo (g x2 z + g y y2)

L5711

Hypothesis H25 : SNo (g x2 y2)

L5712

Hypothesis H26 : SNo (v + g x2 y2)

L5713

Theorem. (Conj_mul_SNo_assoc_lem2__37__12)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is a8f005... and proposition id is b1bb07...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__37__14

L5719

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__37__14 TMGs5bsCzmo4wNE1QdfyoxjsJPcUcezEaMX bounty of about 25 bars ***)

L5720

Variable x : set

L5721

Variable y : set

L5722

Variable z : set

L5723

Variable w : set

L5724

Variable u : set

L5725

Variable v : set

L5726

Variable x2 : set

L5727

Variable y2 : set

L5728

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5729

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5730

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5731

Hypothesis H3 : SNo x

L5732

Hypothesis H4 : SNo y

L5733

Hypothesis H5 : SNo z

L5734

Hypothesis H6 : SNo (g x y)

L5735

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5736

Hypothesis H8 : u ∈ SNoR x

L5737

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5738

Hypothesis H10 : SNo u

L5739

Hypothesis H11 : x < u

L5740

Hypothesis H12 : SNo v

L5741

Hypothesis H13 : SNo (g u v)

L5742

Hypothesis H15 : y2 ∈ SNoR z

L5743

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5744

Hypothesis H17 : SNo x2

L5745

Hypothesis H18 : y < x2

L5746

Hypothesis H19 : SNo y2

L5747

Hypothesis H20 : z < y2

L5748

Hypothesis H21 : SNo (g u y)

L5749

Hypothesis H22 : SNo (g u x2)

L5750

Hypothesis H23 : SNo (g x x2)

L5751

Hypothesis H24 : SNo (g x2 z + g y y2)

L5752

Hypothesis H25 : SNo (g x2 y2)

L5753

Hypothesis H26 : SNo (v + g x2 y2)

L5754

Theorem. (Conj_mul_SNo_assoc_lem2__37__14)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 4ffc39... and proposition id is a29391...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__39__13

L5760

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__39__13 TMNC1VHTJQsNSPKeN389Mcbj6H3V2wfkoF4 bounty of about 25 bars ***)

L5761

Variable x : set

L5762

Variable y : set

L5763

Variable z : set

L5764

Variable w : set

L5765

Variable u : set

L5766

Variable v : set

L5767

Variable x2 : set

L5768

Variable y2 : set

L5769

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5770

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5771

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5772

Hypothesis H3 : SNo x

L5773

Hypothesis H4 : SNo y

L5774

Hypothesis H5 : SNo z

L5775

Hypothesis H6 : SNo (g x y)

L5776

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5777

Hypothesis H8 : u ∈ SNoR x

L5778

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5779

Hypothesis H10 : SNo u

L5780

Hypothesis H11 : x < u

L5781

Hypothesis H12 : SNo v

L5782

Hypothesis H14 : x2 ∈ SNoR y

L5783

Hypothesis H15 : y2 ∈ SNoR z

L5784

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5785

Hypothesis H17 : SNo x2

L5786

Hypothesis H18 : y < x2

L5787

Hypothesis H19 : SNo y2

L5788

Hypothesis H20 : z < y2

L5789

Hypothesis H21 : SNo (g u y)

L5790

Hypothesis H22 : SNo (g u x2)

L5791

Hypothesis H23 : SNo (g x x2)

L5792

Hypothesis H24 : SNo (g x2 z + g y y2)

L5793

Theorem. (Conj_mul_SNo_assoc_lem2__39__13)

SNo (g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 9d01ff... and proposition id is a887aa...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__40__4

L5799

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__40__4 TMb6a9MwoRRTArjL1znmzoSZHREhbRHj9Lo bounty of about 25 bars ***)

L5800

Variable x : set

L5801

Variable y : set

L5802

Variable z : set

L5803

Variable w : set

L5804

Variable u : set

L5805

Variable v : set

L5806

Variable x2 : set

L5807

Variable y2 : set

L5808

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5809

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5810

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5811

Hypothesis H3 : SNo x

L5812

Hypothesis H5 : SNo z

L5813

Hypothesis H6 : SNo (g x y)

L5814

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5815

Hypothesis H8 : u ∈ SNoR x

L5816

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5817

Hypothesis H10 : SNo u

L5818

Hypothesis H11 : x < u

L5819

Hypothesis H12 : SNo v

L5820

Hypothesis H13 : SNo (g u v)

L5821

Hypothesis H14 : x2 ∈ SNoR y

L5822

Hypothesis H15 : y2 ∈ SNoR z

L5823

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5824

Hypothesis H17 : SNo x2

L5825

Hypothesis H18 : y < x2

L5826

Hypothesis H19 : SNo y2

L5827

Hypothesis H20 : z < y2

L5828

Hypothesis H21 : SNo (g u y)

L5829

Hypothesis H22 : SNo (g u x2)

L5830

Hypothesis H23 : SNo (g x x2)

L5831

Hypothesis H24 : SNo (g x2 z)

L5832

Hypothesis H25 : SNo (g y y2)

L5833

Theorem. (Conj_mul_SNo_assoc_lem2__40__4)

SNo (g x2 z + g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is f04c00... and proposition id is 39ff18...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__40__14

L5839

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__40__14 TMHqvQzY4vjQQMo2kF1DN8JZhCa5nHZ98W9 bounty of about 25 bars ***)

L5840

Variable x : set

L5841

Variable y : set

L5842

Variable z : set

L5843

Variable w : set

L5844

Variable u : set

L5845

Variable v : set

L5846

Variable x2 : set

L5847

Variable y2 : set

L5848

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5849

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5850

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5851

Hypothesis H3 : SNo x

L5852

Hypothesis H4 : SNo y

L5853

Hypothesis H5 : SNo z

L5854

Hypothesis H6 : SNo (g x y)

L5855

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5856

Hypothesis H8 : u ∈ SNoR x

L5857

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5858

Hypothesis H10 : SNo u

L5859

Hypothesis H11 : x < u

L5860

Hypothesis H12 : SNo v

L5861

Hypothesis H13 : SNo (g u v)

L5862

Hypothesis H15 : y2 ∈ SNoR z

L5863

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5864

Hypothesis H17 : SNo x2

L5865

Hypothesis H18 : y < x2

L5866

Hypothesis H19 : SNo y2

L5867

Hypothesis H20 : z < y2

L5868

Hypothesis H21 : SNo (g u y)

L5869

Hypothesis H22 : SNo (g u x2)

L5870

Hypothesis H23 : SNo (g x x2)

L5871

Hypothesis H24 : SNo (g x2 z)

L5872

Hypothesis H25 : SNo (g y y2)

L5873

Theorem. (Conj_mul_SNo_assoc_lem2__40__14)

SNo (g x2 z + g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is dc89cc... and proposition id is def2ab...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__42__4

L5879

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__42__4 TMWaixchB1EoAuxtMZyCM8h4swsxiDand6m bounty of about 25 bars ***)

L5880

Variable x : set

L5881

Variable y : set

L5882

Variable z : set

L5883

Variable w : set

L5884

Variable u : set

L5885

Variable v : set

L5886

Variable x2 : set

L5887

Variable y2 : set

L5888

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5889

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5890

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5891

Hypothesis H3 : SNo x

L5892

Hypothesis H5 : SNo z

L5893

Hypothesis H6 : SNo (g x y)

L5894

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5895

Hypothesis H8 : u ∈ SNoR x

L5896

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5897

Hypothesis H10 : SNo u

L5898

Hypothesis H11 : x < u

L5899

Hypothesis H12 : SNo v

L5900

Hypothesis H13 : SNo (g u v)

L5901

Hypothesis H14 : x2 ∈ SNoR y

L5902

Hypothesis H15 : y2 ∈ SNoR z

L5903

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5904

Hypothesis H17 : SNo x2

L5905

Hypothesis H18 : y < x2

L5906

Hypothesis H19 : SNo y2

L5907

Hypothesis H20 : z < y2

L5908

Hypothesis H21 : SNo (g u y)

L5909

Hypothesis H22 : SNo (g u x2)

L5910

Hypothesis H23 : SNo (g x x2)

L5911

Theorem. (Conj_mul_SNo_assoc_lem2__42__4)

SNo (g x2 z) → g (g x y) z < w

In Proofgold the corresponding term root is e061f3... and proposition id is 8a115e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__42__15

L5917

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__42__15 TMYoHo4haydzNdVTLYhr2PP71XFV6f7zJia bounty of about 25 bars ***)

L5918

Variable x : set

L5919

Variable y : set

L5920

Variable z : set

L5921

Variable w : set

L5922

Variable u : set

L5923

Variable v : set

L5924

Variable x2 : set

L5925

Variable y2 : set

L5926

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5927

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5928

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5929

Hypothesis H3 : SNo x

L5930

Hypothesis H4 : SNo y

L5931

Hypothesis H5 : SNo z

L5932

Hypothesis H6 : SNo (g x y)

L5933

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5934

Hypothesis H8 : u ∈ SNoR x

L5935

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5936

Hypothesis H10 : SNo u

L5937

Hypothesis H11 : x < u

L5938

Hypothesis H12 : SNo v

L5939

Hypothesis H13 : SNo (g u v)

L5940

Hypothesis H14 : x2 ∈ SNoR y

L5941

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5942

Hypothesis H17 : SNo x2

L5943

Hypothesis H18 : y < x2

L5944

Hypothesis H19 : SNo y2

L5945

Hypothesis H20 : z < y2

L5946

Hypothesis H21 : SNo (g u y)

L5947

Hypothesis H22 : SNo (g u x2)

L5948

Hypothesis H23 : SNo (g x x2)

L5949

Theorem. (Conj_mul_SNo_assoc_lem2__42__15)

SNo (g x2 z) → g (g x y) z < w

In Proofgold the corresponding term root is b4968d... and proposition id is ca9725...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__42__18

L5955

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__42__18 TMW4eLZAbFu5ejikBcoUuYLMU3As4Zh62VS bounty of about 25 bars ***)

L5956

Variable x : set

L5957

Variable y : set

L5958

Variable z : set

L5959

Variable w : set

L5960

Variable u : set

L5961

Variable v : set

L5962

Variable x2 : set

L5963

Variable y2 : set

L5964

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L5965

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L5966

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L5967

Hypothesis H3 : SNo x

L5968

Hypothesis H4 : SNo y

L5969

Hypothesis H5 : SNo z

L5970

Hypothesis H6 : SNo (g x y)

L5971

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L5972

Hypothesis H8 : u ∈ SNoR x

L5973

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L5974

Hypothesis H10 : SNo u

L5975

Hypothesis H11 : x < u

L5976

Hypothesis H12 : SNo v

L5977

Hypothesis H13 : SNo (g u v)

L5978

Hypothesis H14 : x2 ∈ SNoR y

L5979

Hypothesis H15 : y2 ∈ SNoR z

L5980

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L5981

Hypothesis H17 : SNo x2

L5982

Hypothesis H19 : SNo y2

L5983

Hypothesis H20 : z < y2

L5984

Hypothesis H21 : SNo (g u y)

L5985

Hypothesis H22 : SNo (g u x2)

L5986

Hypothesis H23 : SNo (g x x2)

L5987

Theorem. (Conj_mul_SNo_assoc_lem2__42__18)

SNo (g x2 z) → g (g x y) z < w

In Proofgold the corresponding term root is e257b2... and proposition id is 558af0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__45__4

L5993

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__45__4 TMKdP4SL5ucMoESxPitXVtX2ALgUWHuyNfL bounty of about 25 bars ***)

L5994

Variable x : set

L5995

Variable y : set

L5996

Variable z : set

L5997

Variable w : set

L5998

Variable u : set

L5999

Variable v : set

L6000

Variable x2 : set

L6001

Variable y2 : set

L6002

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6003

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6004

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6005

Hypothesis H3 : SNo x

L6006

Hypothesis H5 : SNo z

L6007

Hypothesis H6 : SNo (g x y)

L6008

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6009

Hypothesis H8 : u ∈ SNoR x

L6010

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6011

Hypothesis H10 : SNo u

L6012

Hypothesis H11 : x < u

L6013

Hypothesis H12 : SNo v

L6014

Hypothesis H13 : SNo (g u v)

L6015

Hypothesis H14 : x2 ∈ SNoR y

L6016

Hypothesis H15 : y2 ∈ SNoR z

L6017

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6018

Hypothesis H17 : SNo x2

L6019

Hypothesis H18 : y < x2

L6020

Hypothesis H19 : SNo y2

L6021

Hypothesis H20 : z < y2

L6022

Theorem. (Conj_mul_SNo_assoc_lem2__45__4)

SNo (g u y) → g (g x y) z < w

In Proofgold the corresponding term root is f88b1e... and proposition id is bf072a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__45__15

L6028

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__45__15 TMTikR4XceorDfdtQiHETfDdabKsUUPvtVy bounty of about 25 bars ***)

L6029

Variable x : set

L6030

Variable y : set

L6031

Variable z : set

L6032

Variable w : set

L6033

Variable u : set

L6034

Variable v : set

L6035

Variable x2 : set

L6036

Variable y2 : set

L6037

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6038

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6039

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6040

Hypothesis H3 : SNo x

L6041

Hypothesis H4 : SNo y

L6042

Hypothesis H5 : SNo z

L6043

Hypothesis H6 : SNo (g x y)

L6044

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6045

Hypothesis H8 : u ∈ SNoR x

L6046

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6047

Hypothesis H10 : SNo u

L6048

Hypothesis H11 : x < u

L6049

Hypothesis H12 : SNo v

L6050

Hypothesis H13 : SNo (g u v)

L6051

Hypothesis H14 : x2 ∈ SNoR y

L6052

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6053

Hypothesis H17 : SNo x2

L6054

Hypothesis H18 : y < x2

L6055

Hypothesis H19 : SNo y2

L6056

Hypothesis H20 : z < y2

L6057

Theorem. (Conj_mul_SNo_assoc_lem2__45__15)

SNo (g u y) → g (g x y) z < w

In Proofgold the corresponding term root is 7e07c8... and proposition id is 3d155c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__48__14

L6063

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__48__14 TMT3nUxzqDef7tfkgrUPSGgMb2ixybp6r6U bounty of about 25 bars ***)

L6064

Variable x : set

L6065

Variable y : set

L6066

Variable z : set

L6067

Variable w : set

L6068

Variable u : set

L6069

Variable v : set

L6070

Variable x2 : set

L6071

Variable y2 : set

L6072

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6073

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6074

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6075

Hypothesis H3 : SNo x

L6076

Hypothesis H4 : SNo y

L6077

Hypothesis H5 : SNo z

L6078

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6079

Hypothesis H7 : u ∈ SNoR x

L6080

Hypothesis H8 : w = g u (g y z) + g x v + - (g u v)

L6081

Hypothesis H9 : SNo u

L6082

Hypothesis H10 : x < u

L6083

Hypothesis H11 : SNo v

L6084

Hypothesis H12 : x2 ∈ SNoL y

L6085

Hypothesis H13 : y2 ∈ SNoL z

L6086

Hypothesis H15 : SNo x2

L6087

Hypothesis H16 : x2 < y

L6088

Hypothesis H17 : SNo y2

L6089

Hypothesis H18 : y2 < z

L6090

Hypothesis H19 : SNo (g x2 z + g y y2)

L6091

Hypothesis H20 : SNo (v + g x2 y2)

L6092

Hypothesis H21 : SNo (g u y + g x x2)

L6093

Hypothesis H22 : SNo (g x y + g u x2)

L6094

Theorem. (Conj_mul_SNo_assoc_lem2__48__14)

SNo (g (g x y + g u x2) z) → g (g x y) z < w

In Proofgold the corresponding term root is 80389c... and proposition id is b7becc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__48__21

L6100

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__48__21 TMMG8gnjS3XAQXfWgCnazW7AkzsvR4uqHKf bounty of about 25 bars ***)

L6101

Variable x : set

L6102

Variable y : set

L6103

Variable z : set

L6104

Variable w : set

L6105

Variable u : set

L6106

Variable v : set

L6107

Variable x2 : set

L6108

Variable y2 : set

L6109

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6110

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6111

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6112

Hypothesis H3 : SNo x

L6113

Hypothesis H4 : SNo y

L6114

Hypothesis H5 : SNo z

L6115

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6116

Hypothesis H7 : u ∈ SNoR x

L6117

Hypothesis H8 : w = g u (g y z) + g x v + - (g u v)

L6118

Hypothesis H9 : SNo u

L6119

Hypothesis H10 : x < u

L6120

Hypothesis H11 : SNo v

L6121

Hypothesis H12 : x2 ∈ SNoL y

L6122

Hypothesis H13 : y2 ∈ SNoL z

L6123

Hypothesis H14 : (v + g x2 y2) ≤ g x2 z + g y y2

L6124

Hypothesis H15 : SNo x2

L6125

Hypothesis H16 : x2 < y

L6126

Hypothesis H17 : SNo y2

L6127

Hypothesis H18 : y2 < z

L6128

Hypothesis H19 : SNo (g x2 z + g y y2)

L6129

Hypothesis H20 : SNo (v + g x2 y2)

L6130

Hypothesis H22 : SNo (g x y + g u x2)

L6131

Theorem. (Conj_mul_SNo_assoc_lem2__48__21)

SNo (g (g x y + g u x2) z) → g (g x y) z < w

In Proofgold the corresponding term root is e7851c... and proposition id is b018ad...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__50__1

L6137

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__50__1 TMR9dqsch3YvE3fTz13KSmSGTddFonY238r bounty of about 25 bars ***)

L6138

Variable x : set

L6139

Variable y : set

L6140

Variable z : set

L6141

Variable w : set

L6142

Variable u : set

L6143

Variable v : set

L6144

Variable x2 : set

L6145

Variable y2 : set

L6146

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6147

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6148

Hypothesis H3 : SNo x

L6149

Hypothesis H4 : SNo y

L6150

Hypothesis H5 : SNo z

L6151

Hypothesis H6 : SNo (g x y)

L6152

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6153

Hypothesis H8 : u ∈ SNoR x

L6154

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6155

Hypothesis H10 : SNo u

L6156

Hypothesis H11 : x < u

L6157

Hypothesis H12 : SNo v

L6158

Hypothesis H13 : x2 ∈ SNoL y

L6159

Hypothesis H14 : y2 ∈ SNoL z

L6160

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L6161

Hypothesis H16 : SNo x2

L6162

Hypothesis H17 : x2 < y

L6163

Hypothesis H18 : SNo y2

L6164

Hypothesis H19 : y2 < z

L6165

Hypothesis H20 : SNo (g u x2)

L6166

Hypothesis H21 : SNo (g x2 z + g y y2)

L6167

Hypothesis H22 : SNo (v + g x2 y2)

L6168

Hypothesis H23 : SNo (g u y + g x x2)

L6169

Theorem. (Conj_mul_SNo_assoc_lem2__50__1)

SNo (g x y + g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is 2e96f5... and proposition id is 6cc9a0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__50__2

L6175

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__50__2 TMWXFjWUL1SKTfJEyohLEaaCkjkSSa2VZSZ bounty of about 25 bars ***)

L6176

Variable x : set

L6177

Variable y : set

L6178

Variable z : set

L6179

Variable w : set

L6180

Variable u : set

L6181

Variable v : set

L6182

Variable x2 : set

L6183

Variable y2 : set

L6184

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6185

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6186

Hypothesis H3 : SNo x

L6187

Hypothesis H4 : SNo y

L6188

Hypothesis H5 : SNo z

L6189

Hypothesis H6 : SNo (g x y)

L6190

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6191

Hypothesis H8 : u ∈ SNoR x

L6192

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6193

Hypothesis H10 : SNo u

L6194

Hypothesis H11 : x < u

L6195

Hypothesis H12 : SNo v

L6196

Hypothesis H13 : x2 ∈ SNoL y

L6197

Hypothesis H14 : y2 ∈ SNoL z

L6198

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L6199

Hypothesis H16 : SNo x2

L6200

Hypothesis H17 : x2 < y

L6201

Hypothesis H18 : SNo y2

L6202

Hypothesis H19 : y2 < z

L6203

Hypothesis H20 : SNo (g u x2)

L6204

Hypothesis H21 : SNo (g x2 z + g y y2)

L6205

Hypothesis H22 : SNo (v + g x2 y2)

L6206

Hypothesis H23 : SNo (g u y + g x x2)

L6207

Theorem. (Conj_mul_SNo_assoc_lem2__50__2)

SNo (g x y + g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is 5726a2... and proposition id is 3368ed...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__51__12

L6213

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__51__12 TMFHLmKnnfKs1D3E6NyWmmWRaZYNxf5Gfcw bounty of about 25 bars ***)

L6214

Variable x : set

L6215

Variable y : set

L6216

Variable z : set

L6217

Variable w : set

L6218

Variable u : set

L6219

Variable v : set

L6220

Variable x2 : set

L6221

Variable y2 : set

L6222

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6223

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6224

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6225

Hypothesis H3 : SNo x

L6226

Hypothesis H4 : SNo y

L6227

Hypothesis H5 : SNo z

L6228

Hypothesis H6 : SNo (g x y)

L6229

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6230

Hypothesis H8 : u ∈ SNoR x

L6231

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6232

Hypothesis H10 : SNo u

L6233

Hypothesis H11 : x < u

L6234

Hypothesis H13 : x2 ∈ SNoL y

L6235

Hypothesis H14 : y2 ∈ SNoL z

L6236

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L6237

Hypothesis H16 : SNo x2

L6238

Hypothesis H17 : x2 < y

L6239

Hypothesis H18 : SNo y2

L6240

Hypothesis H19 : y2 < z

L6241

Hypothesis H20 : SNo (g u y)

L6242

Hypothesis H21 : SNo (g u x2)

L6243

Hypothesis H22 : SNo (g x x2)

L6244

Hypothesis H23 : SNo (g x2 z + g y y2)

L6245

Hypothesis H24 : SNo (v + g x2 y2)

L6246

Theorem. (Conj_mul_SNo_assoc_lem2__51__12)

SNo (g u y + g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 5b4bcb... and proposition id is e6a7e1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__51__15

L6252

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__51__15 TMQHV9L2FKs5b2X4EeX1xboocv4thgkmLHx bounty of about 25 bars ***)

L6253

Variable x : set

L6254

Variable y : set

L6255

Variable z : set

L6256

Variable w : set

L6257

Variable u : set

L6258

Variable v : set

L6259

Variable x2 : set

L6260

Variable y2 : set

L6261

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6262

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6263

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6264

Hypothesis H3 : SNo x

L6265

Hypothesis H4 : SNo y

L6266

Hypothesis H5 : SNo z

L6267

Hypothesis H6 : SNo (g x y)

L6268

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6269

Hypothesis H8 : u ∈ SNoR x

L6270

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6271

Hypothesis H10 : SNo u

L6272

Hypothesis H11 : x < u

L6273

Hypothesis H12 : SNo v

L6274

Hypothesis H13 : x2 ∈ SNoL y

L6275

Hypothesis H14 : y2 ∈ SNoL z

L6276

Hypothesis H16 : SNo x2

L6277

Hypothesis H17 : x2 < y

L6278

Hypothesis H18 : SNo y2

L6279

Hypothesis H19 : y2 < z

L6280

Hypothesis H20 : SNo (g u y)

L6281

Hypothesis H21 : SNo (g u x2)

L6282

Hypothesis H22 : SNo (g x x2)

L6283

Hypothesis H23 : SNo (g x2 z + g y y2)

L6284

Hypothesis H24 : SNo (v + g x2 y2)

L6285

Theorem. (Conj_mul_SNo_assoc_lem2__51__15)

SNo (g u y + g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 3ec45c... and proposition id is f03cba...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__52__12

L6291

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__52__12 TMZbKWyWifTHPMaSivPiEPtwfYMUrTPqcTL bounty of about 25 bars ***)

L6292

Variable x : set

L6293

Variable y : set

L6294

Variable z : set

L6295

Variable w : set

L6296

Variable u : set

L6297

Variable v : set

L6298

Variable x2 : set

L6299

Variable y2 : set

L6300

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6301

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6302

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6303

Hypothesis H3 : SNo x

L6304

Hypothesis H4 : SNo y

L6305

Hypothesis H5 : SNo z

L6306

Hypothesis H6 : SNo (g x y)

L6307

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6308

Hypothesis H8 : u ∈ SNoR x

L6309

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6310

Hypothesis H10 : SNo u

L6311

Hypothesis H11 : x < u

L6312

Hypothesis H13 : x2 ∈ SNoL y

L6313

Hypothesis H14 : y2 ∈ SNoL z

L6314

Hypothesis H15 : (v + g x2 y2) ≤ g x2 z + g y y2

L6315

Hypothesis H16 : SNo x2

L6316

Hypothesis H17 : x2 < y

L6317

Hypothesis H18 : SNo y2

L6318

Hypothesis H19 : y2 < z

L6319

Hypothesis H20 : SNo (g u y)

L6320

Hypothesis H21 : SNo (g u x2)

L6321

Hypothesis H22 : SNo (g x x2)

L6322

Hypothesis H23 : SNo (g x2 z + g y y2)

L6323

Hypothesis H24 : SNo (v + g x2 y2)

L6324

Theorem. (Conj_mul_SNo_assoc_lem2__52__12)

SNo (g x (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is e5308e... and proposition id is b1407e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__53__6

L6330

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__53__6 TMcBPERf69M1tWrG1ru9KxKcaaKUTL15Rf9 bounty of about 25 bars ***)

L6331

Variable x : set

L6332

Variable y : set

L6333

Variable z : set

L6334

Variable w : set

L6335

Variable u : set

L6336

Variable v : set

L6337

Variable x2 : set

L6338

Variable y2 : set

L6339

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6340

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6341

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6342

Hypothesis H3 : SNo x

L6343

Hypothesis H4 : SNo y

L6344

Hypothesis H5 : SNo z

L6345

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6346

Hypothesis H8 : u ∈ SNoR x

L6347

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6348

Hypothesis H10 : SNo u

L6349

Hypothesis H11 : x < u

L6350

Hypothesis H12 : SNo v

L6351

Hypothesis H13 : SNo (g u v)

L6352

Hypothesis H14 : x2 ∈ SNoL y

L6353

Hypothesis H15 : y2 ∈ SNoL z

L6354

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6355

Hypothesis H17 : SNo x2

L6356

Hypothesis H18 : x2 < y

L6357

Hypothesis H19 : SNo y2

L6358

Hypothesis H20 : y2 < z

L6359

Hypothesis H21 : SNo (g u y)

L6360

Hypothesis H22 : SNo (g u x2)

L6361

Hypothesis H23 : SNo (g x x2)

L6362

Hypothesis H24 : SNo (g x2 z + g y y2)

L6363

Hypothesis H25 : SNo (v + g x2 y2)

L6364

Hypothesis H26 : SNo (g u (g x2 y2))

L6365

Theorem. (Conj_mul_SNo_assoc_lem2__53__6)

SNo (g u v + g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 6901b9... and proposition id is cb3108...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__53__18

L6371

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__53__18 TMMMviAFZ4dJ423MZrq7U5TYpiHh8teJZrq bounty of about 25 bars ***)

L6372

Variable x : set

L6373

Variable y : set

L6374

Variable z : set

L6375

Variable w : set

L6376

Variable u : set

L6377

Variable v : set

L6378

Variable x2 : set

L6379

Variable y2 : set

L6380

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6381

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6382

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6383

Hypothesis H3 : SNo x

L6384

Hypothesis H4 : SNo y

L6385

Hypothesis H5 : SNo z

L6386

Hypothesis H6 : SNo (g x y)

L6387

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6388

Hypothesis H8 : u ∈ SNoR x

L6389

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6390

Hypothesis H10 : SNo u

L6391

Hypothesis H11 : x < u

L6392

Hypothesis H12 : SNo v

L6393

Hypothesis H13 : SNo (g u v)

L6394

Hypothesis H14 : x2 ∈ SNoL y

L6395

Hypothesis H15 : y2 ∈ SNoL z

L6396

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6397

Hypothesis H17 : SNo x2

L6398

Hypothesis H19 : SNo y2

L6399

Hypothesis H20 : y2 < z

L6400

Hypothesis H21 : SNo (g u y)

L6401

Hypothesis H22 : SNo (g u x2)

L6402

Hypothesis H23 : SNo (g x x2)

L6403

Hypothesis H24 : SNo (g x2 z + g y y2)

L6404

Hypothesis H25 : SNo (v + g x2 y2)

L6405

Hypothesis H26 : SNo (g u (g x2 y2))

L6406

Theorem. (Conj_mul_SNo_assoc_lem2__53__18)

SNo (g u v + g u (g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is adfa39... and proposition id is faca17...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__54__10

L6412

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__54__10 TMKDAAdxAjZFsnpWNqph7UBxBfXqJRq8vHf bounty of about 25 bars ***)

L6413

Variable x : set

L6414

Variable y : set

L6415

Variable z : set

L6416

Variable w : set

L6417

Variable u : set

L6418

Variable v : set

L6419

Variable x2 : set

L6420

Variable y2 : set

L6421

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6422

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6423

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6424

Hypothesis H3 : SNo x

L6425

Hypothesis H4 : SNo y

L6426

Hypothesis H5 : SNo z

L6427

Hypothesis H6 : SNo (g x y)

L6428

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6429

Hypothesis H8 : u ∈ SNoR x

L6430

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6431

Hypothesis H11 : x < u

L6432

Hypothesis H12 : SNo v

L6433

Hypothesis H13 : SNo (g u v)

L6434

Hypothesis H14 : x2 ∈ SNoL y

L6435

Hypothesis H15 : y2 ∈ SNoL z

L6436

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6437

Hypothesis H17 : SNo x2

L6438

Hypothesis H18 : x2 < y

L6439

Hypothesis H19 : SNo y2

L6440

Hypothesis H20 : y2 < z

L6441

Hypothesis H21 : SNo (g u y)

L6442

Hypothesis H22 : SNo (g u x2)

L6443

Hypothesis H23 : SNo (g x x2)

L6444

Hypothesis H24 : SNo (g x2 z + g y y2)

L6445

Hypothesis H25 : SNo (v + g x2 y2)

L6446

Hypothesis H26 : SNo (g u (g x2 y2))

L6447

Theorem. (Conj_mul_SNo_assoc_lem2__54__10)

SNo (g u (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 43c84e... and proposition id is ea6845...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__57__11

L6453

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__57__11 TMQdNjtjCmVVxgeAwdQ242WWkn7SG6zFt1A bounty of about 25 bars ***)

L6454

Variable x : set

L6455

Variable y : set

L6456

Variable z : set

L6457

Variable w : set

L6458

Variable u : set

L6459

Variable v : set

L6460

Variable x2 : set

L6461

Variable y2 : set

L6462

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6463

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6464

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6465

Hypothesis H3 : SNo x

L6466

Hypothesis H4 : SNo y

L6467

Hypothesis H5 : SNo z

L6468

Hypothesis H6 : SNo (g x y)

L6469

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6470

Hypothesis H8 : u ∈ SNoR x

L6471

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6472

Hypothesis H10 : SNo u

L6473

Hypothesis H12 : SNo v

L6474

Hypothesis H13 : SNo (g u v)

L6475

Hypothesis H14 : x2 ∈ SNoL y

L6476

Hypothesis H15 : y2 ∈ SNoL z

L6477

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6478

Hypothesis H17 : SNo x2

L6479

Hypothesis H18 : x2 < y

L6480

Hypothesis H19 : SNo y2

L6481

Hypothesis H20 : y2 < z

L6482

Hypothesis H21 : SNo (g u y)

L6483

Hypothesis H22 : SNo (g u x2)

L6484

Hypothesis H23 : SNo (g x x2)

L6485

Hypothesis H24 : SNo (g x2 z + g y y2)

L6486

Hypothesis H25 : SNo (g x2 y2)

L6487

Hypothesis H26 : SNo (v + g x2 y2)

L6488

Theorem. (Conj_mul_SNo_assoc_lem2__57__11)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 8890dd... and proposition id is d47cdc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__57__18

L6494

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__57__18 TMdeaupC9Jes2pmEFaJHftNLt2N6ivEbMfu bounty of about 25 bars ***)

L6495

Variable x : set

L6496

Variable y : set

L6497

Variable z : set

L6498

Variable w : set

L6499

Variable u : set

L6500

Variable v : set

L6501

Variable x2 : set

L6502

Variable y2 : set

L6503

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6504

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6505

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6506

Hypothesis H3 : SNo x

L6507

Hypothesis H4 : SNo y

L6508

Hypothesis H5 : SNo z

L6509

Hypothesis H6 : SNo (g x y)

L6510

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6511

Hypothesis H8 : u ∈ SNoR x

L6512

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6513

Hypothesis H10 : SNo u

L6514

Hypothesis H11 : x < u

L6515

Hypothesis H12 : SNo v

L6516

Hypothesis H13 : SNo (g u v)

L6517

Hypothesis H14 : x2 ∈ SNoL y

L6518

Hypothesis H15 : y2 ∈ SNoL z

L6519

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6520

Hypothesis H17 : SNo x2

L6521

Hypothesis H19 : SNo y2

L6522

Hypothesis H20 : y2 < z

L6523

Hypothesis H21 : SNo (g u y)

L6524

Hypothesis H22 : SNo (g u x2)

L6525

Hypothesis H23 : SNo (g x x2)

L6526

Hypothesis H24 : SNo (g x2 z + g y y2)

L6527

Hypothesis H25 : SNo (g x2 y2)

L6528

Hypothesis H26 : SNo (v + g x2 y2)

L6529

Theorem. (Conj_mul_SNo_assoc_lem2__57__18)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 9e7857... and proposition id is 0ea3fd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__58__3

L6535

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__58__3 TMKLqfPLWB6RGNvuGUYLVsanWVM8j9PY7ij bounty of about 25 bars ***)

L6536

Variable x : set

L6537

Variable y : set

L6538

Variable z : set

L6539

Variable w : set

L6540

Variable u : set

L6541

Variable v : set

L6542

Variable x2 : set

L6543

Variable y2 : set

L6544

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6545

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6546

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6547

Hypothesis H4 : SNo y

L6548

Hypothesis H5 : SNo z

L6549

Hypothesis H6 : SNo (g x y)

L6550

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6551

Hypothesis H8 : u ∈ SNoR x

L6552

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6553

Hypothesis H10 : SNo u

L6554

Hypothesis H11 : x < u

L6555

Hypothesis H12 : SNo v

L6556

Hypothesis H13 : SNo (g u v)

L6557

Hypothesis H14 : x2 ∈ SNoL y

L6558

Hypothesis H15 : y2 ∈ SNoL z

L6559

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6560

Hypothesis H17 : SNo x2

L6561

Hypothesis H18 : x2 < y

L6562

Hypothesis H19 : SNo y2

L6563

Hypothesis H20 : y2 < z

L6564

Hypothesis H21 : SNo (g u y)

L6565

Hypothesis H22 : SNo (g u x2)

L6566

Hypothesis H23 : SNo (g x x2)

L6567

Hypothesis H24 : SNo (g x2 z + g y y2)

L6568

Hypothesis H25 : SNo (g x2 y2)

L6569

Theorem. (Conj_mul_SNo_assoc_lem2__58__3)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is e7c58e... and proposition id is 9a1ecd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__59__19

L6575

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__59__19 TMF1G8GFaQpLFVC51BLDZNtsA3SvTuw4C7V bounty of about 25 bars ***)

L6576

Variable x : set

L6577

Variable y : set

L6578

Variable z : set

L6579

Variable w : set

L6580

Variable u : set

L6581

Variable v : set

L6582

Variable x2 : set

L6583

Variable y2 : set

L6584

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6585

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6586

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6587

Hypothesis H3 : SNo x

L6588

Hypothesis H4 : SNo y

L6589

Hypothesis H5 : SNo z

L6590

Hypothesis H6 : SNo (g x y)

L6591

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6592

Hypothesis H8 : u ∈ SNoR x

L6593

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6594

Hypothesis H10 : SNo u

L6595

Hypothesis H11 : x < u

L6596

Hypothesis H12 : SNo v

L6597

Hypothesis H13 : SNo (g u v)

L6598

Hypothesis H14 : x2 ∈ SNoL y

L6599

Hypothesis H15 : y2 ∈ SNoL z

L6600

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6601

Hypothesis H17 : SNo x2

L6602

Hypothesis H18 : x2 < y

L6603

Hypothesis H20 : y2 < z

L6604

Hypothesis H21 : SNo (g u y)

L6605

Hypothesis H22 : SNo (g u x2)

L6606

Hypothesis H23 : SNo (g x x2)

L6607

Hypothesis H24 : SNo (g x2 z + g y y2)

L6608

Theorem. (Conj_mul_SNo_assoc_lem2__59__19)

SNo (g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 8f59b4... and proposition id is 182b1d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__60__9

L6614

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__60__9 TMRAzdzCAgW3Dd25Lz6eN9ZHC3TLy6SrZPU bounty of about 25 bars ***)

L6615

Variable x : set

L6616

Variable y : set

L6617

Variable z : set

L6618

Variable w : set

L6619

Variable u : set

L6620

Variable v : set

L6621

Variable x2 : set

L6622

Variable y2 : set

L6623

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6624

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6625

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6626

Hypothesis H3 : SNo x

L6627

Hypothesis H4 : SNo y

L6628

Hypothesis H5 : SNo z

L6629

Hypothesis H6 : SNo (g x y)

L6630

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6631

Hypothesis H8 : u ∈ SNoR x

L6632

Hypothesis H10 : SNo u

L6633

Hypothesis H11 : x < u

L6634

Hypothesis H12 : SNo v

L6635

Hypothesis H13 : SNo (g u v)

L6636

Hypothesis H14 : x2 ∈ SNoL y

L6637

Hypothesis H15 : y2 ∈ SNoL z

L6638

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6639

Hypothesis H17 : SNo x2

L6640

Hypothesis H18 : x2 < y

L6641

Hypothesis H19 : SNo y2

L6642

Hypothesis H20 : y2 < z

L6643

Hypothesis H21 : SNo (g u y)

L6644

Hypothesis H22 : SNo (g u x2)

L6645

Hypothesis H23 : SNo (g x x2)

L6646

Hypothesis H24 : SNo (g x2 z)

L6647

Hypothesis H25 : SNo (g y y2)

L6648

Theorem. (Conj_mul_SNo_assoc_lem2__60__9)

SNo (g x2 z + g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is 8f9131... and proposition id is 0be907...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__60__20

L6654

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__60__20 TMQsuQgfVu1W87P8ZJZt8vr8shcutXVGACx bounty of about 25 bars ***)

L6655

Variable x : set

L6656

Variable y : set

L6657

Variable z : set

L6658

Variable w : set

L6659

Variable u : set

L6660

Variable v : set

L6661

Variable x2 : set

L6662

Variable y2 : set

L6663

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6664

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6665

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6666

Hypothesis H3 : SNo x

L6667

Hypothesis H4 : SNo y

L6668

Hypothesis H5 : SNo z

L6669

Hypothesis H6 : SNo (g x y)

L6670

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6671

Hypothesis H8 : u ∈ SNoR x

L6672

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6673

Hypothesis H10 : SNo u

L6674

Hypothesis H11 : x < u

L6675

Hypothesis H12 : SNo v

L6676

Hypothesis H13 : SNo (g u v)

L6677

Hypothesis H14 : x2 ∈ SNoL y

L6678

Hypothesis H15 : y2 ∈ SNoL z

L6679

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6680

Hypothesis H17 : SNo x2

L6681

Hypothesis H18 : x2 < y

L6682

Hypothesis H19 : SNo y2

L6683

Hypothesis H21 : SNo (g u y)

L6684

Hypothesis H22 : SNo (g u x2)

L6685

Hypothesis H23 : SNo (g x x2)

L6686

Hypothesis H24 : SNo (g x2 z)

L6687

Hypothesis H25 : SNo (g y y2)

L6688

Theorem. (Conj_mul_SNo_assoc_lem2__60__20)

SNo (g x2 z + g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is c41f00... and proposition id is 37b374...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__62__21

L6694

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__62__21 TMdYqkDR1gnzUoaagccMDFBfiBkdqHz725X bounty of about 25 bars ***)

L6695

Variable x : set

L6696

Variable y : set

L6697

Variable z : set

L6698

Variable w : set

L6699

Variable u : set

L6700

Variable v : set

L6701

Variable x2 : set

L6702

Variable y2 : set

L6703

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6704

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6705

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6706

Hypothesis H3 : SNo x

L6707

Hypothesis H4 : SNo y

L6708

Hypothesis H5 : SNo z

L6709

Hypothesis H6 : SNo (g x y)

L6710

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6711

Hypothesis H8 : u ∈ SNoR x

L6712

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6713

Hypothesis H10 : SNo u

L6714

Hypothesis H11 : x < u

L6715

Hypothesis H12 : SNo v

L6716

Hypothesis H13 : SNo (g u v)

L6717

Hypothesis H14 : x2 ∈ SNoL y

L6718

Hypothesis H15 : y2 ∈ SNoL z

L6719

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6720

Hypothesis H17 : SNo x2

L6721

Hypothesis H18 : x2 < y

L6722

Hypothesis H19 : SNo y2

L6723

Hypothesis H20 : y2 < z

L6724

Hypothesis H22 : SNo (g u x2)

L6725

Hypothesis H23 : SNo (g x x2)

L6726

Theorem. (Conj_mul_SNo_assoc_lem2__62__21)

SNo (g x2 z) → g (g x y) z < w

In Proofgold the corresponding term root is 4beefa... and proposition id is 14a1de...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__63__4

L6732

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__63__4 TMZrMdAMJxccKMmWwq8Tf2PjEd1t3PTPneh bounty of about 25 bars ***)

L6733

Variable x : set

L6734

Variable y : set

L6735

Variable z : set

L6736

Variable w : set

L6737

Variable u : set

L6738

Variable v : set

L6739

Variable x2 : set

L6740

Variable y2 : set

L6741

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6742

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6743

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6744

Hypothesis H3 : SNo x

L6745

Hypothesis H5 : SNo z

L6746

Hypothesis H6 : SNo (g x y)

L6747

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6748

Hypothesis H8 : u ∈ SNoR x

L6749

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6750

Hypothesis H10 : SNo u

L6751

Hypothesis H11 : x < u

L6752

Hypothesis H12 : SNo v

L6753

Hypothesis H13 : SNo (g u v)

L6754

Hypothesis H14 : x2 ∈ SNoL y

L6755

Hypothesis H15 : y2 ∈ SNoL z

L6756

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6757

Hypothesis H17 : SNo x2

L6758

Hypothesis H18 : x2 < y

L6759

Hypothesis H19 : SNo y2

L6760

Hypothesis H20 : y2 < z

L6761

Hypothesis H21 : SNo (g u y)

L6762

Hypothesis H22 : SNo (g u x2)

L6763

Theorem. (Conj_mul_SNo_assoc_lem2__63__4)

SNo (g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 8709a7... and proposition id is 215877...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__63__5

L6769

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__63__5 TMLxbz6cJ1ydgsj4KDLcfgmWCtRbsM7G6cf bounty of about 25 bars ***)

L6770

Variable x : set

L6771

Variable y : set

L6772

Variable z : set

L6773

Variable w : set

L6774

Variable u : set

L6775

Variable v : set

L6776

Variable x2 : set

L6777

Variable y2 : set

L6778

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6779

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6780

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6781

Hypothesis H3 : SNo x

L6782

Hypothesis H4 : SNo y

L6783

Hypothesis H6 : SNo (g x y)

L6784

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6785

Hypothesis H8 : u ∈ SNoR x

L6786

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6787

Hypothesis H10 : SNo u

L6788

Hypothesis H11 : x < u

L6789

Hypothesis H12 : SNo v

L6790

Hypothesis H13 : SNo (g u v)

L6791

Hypothesis H14 : x2 ∈ SNoL y

L6792

Hypothesis H15 : y2 ∈ SNoL z

L6793

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6794

Hypothesis H17 : SNo x2

L6795

Hypothesis H18 : x2 < y

L6796

Hypothesis H19 : SNo y2

L6797

Hypothesis H20 : y2 < z

L6798

Hypothesis H21 : SNo (g u y)

L6799

Hypothesis H22 : SNo (g u x2)

L6800

Theorem. (Conj_mul_SNo_assoc_lem2__63__5)

SNo (g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 2a52db... and proposition id is 0ac573...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__64__4

L6806

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__64__4 TMWEe4j6FD7anaRNuDY88CfdHkG9mZtWMeU bounty of about 25 bars ***)

L6807

Variable x : set

L6808

Variable y : set

L6809

Variable z : set

L6810

Variable w : set

L6811

Variable u : set

L6812

Variable v : set

L6813

Variable x2 : set

L6814

Variable y2 : set

L6815

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6816

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6817

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6818

Hypothesis H3 : SNo x

L6819

Hypothesis H5 : SNo z

L6820

Hypothesis H6 : SNo (g x y)

L6821

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6822

Hypothesis H8 : u ∈ SNoR x

L6823

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6824

Hypothesis H10 : SNo u

L6825

Hypothesis H11 : x < u

L6826

Hypothesis H12 : SNo v

L6827

Hypothesis H13 : SNo (g u v)

L6828

Hypothesis H14 : x2 ∈ SNoL y

L6829

Hypothesis H15 : y2 ∈ SNoL z

L6830

Hypothesis H16 : (v + g x2 y2) ≤ g x2 z + g y y2

L6831

Hypothesis H17 : SNo x2

L6832

Hypothesis H18 : x2 < y

L6833

Hypothesis H19 : SNo y2

L6834

Hypothesis H20 : y2 < z

L6835

Hypothesis H21 : SNo (g u y)

L6836

Theorem. (Conj_mul_SNo_assoc_lem2__64__4)

SNo (g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is 3d4c93... and proposition id is 11ea5f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__66__10

L6842

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__66__10 TMNjyCWGVtHZxe5RXRaboKnJBQq2mEi3aGr bounty of about 25 bars ***)

L6843

Variable x : set

L6844

Variable y : set

L6845

Variable z : set

L6846

Variable w : set

L6847

Variable u : set

L6848

Variable v : set

L6849

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L6850

Hypothesis H1 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → (∀z2 : set, z2 ∈ SNoL (g x2 y2) → (∀P : prop, (∀w2 : set, w2 ∈ SNoL x2 → (∀u2 : set, u2 ∈ SNoL y2 → (z2 + g w2 u2) ≤ g w2 y2 + g x2 u2 → P)) → (∀w2 : set, w2 ∈ SNoR x2 → (∀u2 : set, u2 ∈ SNoR y2 → (z2 + g w2 u2) ≤ g w2 y2 + g x2 u2 → P)) → P)))

L6851

Hypothesis H2 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L6852

Hypothesis H3 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 ≤ x2 → w2 ≤ y2 → (g z2 y2 + g x2 w2) ≤ g x2 y2 + g z2 w2)

L6853

Hypothesis H4 : SNo x

L6854

Hypothesis H5 : SNo y

L6855

Hypothesis H6 : SNo z

L6856

Hypothesis H7 : SNo (g x y)

L6857

Hypothesis H8 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, SNo y2 → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → w = g x2 (g y z) + g x y2 + - (g x2 y2) → (g x (g z2 z + g y w2) + g x2 (y2 + g z2 w2)) ≤ g x2 (g z2 z + g y w2) + g x (y2 + g z2 w2) → (g (g x y + g x2 z2) z + g (g x2 y + g x z2) w2) < g (g x2 y + g x z2) z + g (g x y + g x2 z2) w2 → g (g x y) z < w))))

L6858

Hypothesis H9 : u ∈ SNoR x

L6859

Hypothesis H11 : w = g u (g y z) + g x v + - (g u v)

L6860

Hypothesis H12 : SNo u

L6861

Hypothesis H13 : x < u

L6862

Hypothesis H14 : SNo v

L6863

Theorem. (Conj_mul_SNo_assoc_lem2__66__10)

SNo (g u v) → g (g x y) z < w

In Proofgold the corresponding term root is 912a50... and proposition id is a60c98...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__71__8

L6869

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__71__8 TMUia5NQQZn5Cv6LDEDti2qfsbYZ9MJQiqP bounty of about 25 bars ***)

L6870

Variable x : set

L6871

Variable y : set

L6872

Variable z : set

L6873

Variable w : set

L6874

Variable u : set

L6875

Variable v : set

L6876

Variable x2 : set

L6877

Variable y2 : set

L6878

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6879

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6880

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6881

Hypothesis H3 : SNo x

L6882

Hypothesis H4 : SNo y

L6883

Hypothesis H5 : SNo z

L6884

Hypothesis H6 : SNo (g x y)

L6885

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6886

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6887

Hypothesis H10 : SNo u

L6888

Hypothesis H11 : u < x

L6889

Hypothesis H12 : SNo v

L6890

Hypothesis H13 : x2 ∈ SNoR y

L6891

Hypothesis H14 : y2 ∈ SNoL z

L6892

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L6893

Hypothesis H16 : SNo x2

L6894

Hypothesis H17 : y < x2

L6895

Hypothesis H18 : SNo y2

L6896

Hypothesis H19 : y2 < z

L6897

Hypothesis H20 : SNo (g u y)

L6898

Hypothesis H21 : SNo (g u x2)

L6899

Hypothesis H22 : SNo (g x x2)

L6900

Hypothesis H23 : SNo (g x2 z + g y y2)

L6901

Hypothesis H24 : SNo (v + g x2 y2)

L6902

Hypothesis H25 : SNo (g u (v + g x2 y2))

L6903

Hypothesis H26 : SNo (g u (g x2 z + g y y2))

L6904

Theorem. (Conj_mul_SNo_assoc_lem2__71__8)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 963450... and proposition id is cf547a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__72__20

L6910

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__72__20 TMF4BUCpvuMHoGCdWRB93SwdFMUk5MSEiEN bounty of about 25 bars ***)

L6911

Variable x : set

L6912

Variable y : set

L6913

Variable z : set

L6914

Variable w : set

L6915

Variable u : set

L6916

Variable v : set

L6917

Variable x2 : set

L6918

Variable y2 : set

L6919

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6920

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6921

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6922

Hypothesis H3 : SNo x

L6923

Hypothesis H4 : SNo y

L6924

Hypothesis H5 : SNo z

L6925

Hypothesis H6 : SNo (g x y)

L6926

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L6927

Hypothesis H8 : u ∈ SNoL x

L6928

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6929

Hypothesis H10 : SNo u

L6930

Hypothesis H11 : u < x

L6931

Hypothesis H12 : SNo v

L6932

Hypothesis H13 : x2 ∈ SNoR y

L6933

Hypothesis H14 : y2 ∈ SNoL z

L6934

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L6935

Hypothesis H16 : SNo x2

L6936

Hypothesis H17 : y < x2

L6937

Hypothesis H18 : SNo y2

L6938

Hypothesis H19 : y2 < z

L6939

Hypothesis H21 : SNo (g u x2)

L6940

Hypothesis H22 : SNo (g x x2)

L6941

Hypothesis H23 : SNo (g x2 z + g y y2)

L6942

Hypothesis H24 : SNo (v + g x2 y2)

L6943

Hypothesis H25 : SNo (g u (v + g x2 y2))

L6944

Theorem. (Conj_mul_SNo_assoc_lem2__72__20)

SNo (g u (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 1f8aa6... and proposition id is a91baf...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__73__7

L6950

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__73__7 TMVRFVxgNYAFtVbcFAr9LAwtaHhGgjw1rfQ bounty of about 25 bars ***)

L6951

Variable x : set

L6952

Variable y : set

L6953

Variable z : set

L6954

Variable w : set

L6955

Variable u : set

L6956

Variable v : set

L6957

Variable x2 : set

L6958

Variable y2 : set

L6959

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6960

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L6961

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L6962

Hypothesis H3 : SNo x

L6963

Hypothesis H4 : SNo y

L6964

Hypothesis H5 : SNo z

L6965

Hypothesis H6 : SNo (g x y)

L6966

Hypothesis H8 : u ∈ SNoL x

L6967

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L6968

Hypothesis H10 : SNo u

L6969

Hypothesis H11 : u < x

L6970

Hypothesis H12 : SNo v

L6971

Hypothesis H13 : x2 ∈ SNoR y

L6972

Hypothesis H14 : y2 ∈ SNoL z

L6973

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L6974

Hypothesis H16 : SNo x2

L6975

Hypothesis H17 : y < x2

L6976

Hypothesis H18 : SNo y2

L6977

Hypothesis H19 : y2 < z

L6978

Hypothesis H20 : SNo (g u y)

L6979

Hypothesis H21 : SNo (g u x2)

L6980

Hypothesis H22 : SNo (g x x2)

L6981

Hypothesis H23 : SNo (g x2 z + g y y2)

L6982

Hypothesis H24 : SNo (v + g x2 y2)

L6983

Theorem. (Conj_mul_SNo_assoc_lem2__73__7)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is f7dce4... and proposition id is 1feda2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__73__10

L6989

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__73__10 TMN4m6Vn2ZqKBY1DaWq3DUNK1aUgTXznV2E bounty of about 25 bars ***)

L6990

Variable x : set

L6991

Variable y : set

L6992

Variable z : set

L6993

Variable w : set

L6994

Variable u : set

L6995

Variable v : set

L6996

Variable x2 : set

L6997

Variable y2 : set

L6998

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L6999

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7000

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7001

Hypothesis H3 : SNo x

L7002

Hypothesis H4 : SNo y

L7003

Hypothesis H5 : SNo z

L7004

Hypothesis H6 : SNo (g x y)

L7005

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7006

Hypothesis H8 : u ∈ SNoL x

L7007

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7008

Hypothesis H11 : u < x

L7009

Hypothesis H12 : SNo v

L7010

Hypothesis H13 : x2 ∈ SNoR y

L7011

Hypothesis H14 : y2 ∈ SNoL z

L7012

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7013

Hypothesis H16 : SNo x2

L7014

Hypothesis H17 : y < x2

L7015

Hypothesis H18 : SNo y2

L7016

Hypothesis H19 : y2 < z

L7017

Hypothesis H20 : SNo (g u y)

L7018

Hypothesis H21 : SNo (g u x2)

L7019

Hypothesis H22 : SNo (g x x2)

L7020

Hypothesis H23 : SNo (g x2 z + g y y2)

L7021

Hypothesis H24 : SNo (v + g x2 y2)

L7022

Theorem. (Conj_mul_SNo_assoc_lem2__73__10)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 12d819... and proposition id is c052cd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__74__16

L7028

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__74__16 TMFpDLWvYyv1SHGThwBjdMe7nTT8c3DHYgm bounty of about 25 bars ***)

L7029

Variable x : set

L7030

Variable y : set

L7031

Variable z : set

L7032

Variable w : set

L7033

Variable u : set

L7034

Variable v : set

L7035

Variable x2 : set

L7036

Variable y2 : set

L7037

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7038

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7039

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7040

Hypothesis H3 : SNo x

L7041

Hypothesis H4 : SNo y

L7042

Hypothesis H5 : SNo z

L7043

Hypothesis H6 : SNo (g x y)

L7044

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7045

Hypothesis H8 : u ∈ SNoL x

L7046

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7047

Hypothesis H10 : SNo u

L7048

Hypothesis H11 : u < x

L7049

Hypothesis H12 : SNo v

L7050

Hypothesis H13 : x2 ∈ SNoR y

L7051

Hypothesis H14 : y2 ∈ SNoL z

L7052

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7053

Hypothesis H17 : y < x2

L7054

Hypothesis H18 : SNo y2

L7055

Hypothesis H19 : y2 < z

L7056

Hypothesis H20 : SNo (g u y)

L7057

Hypothesis H21 : SNo (g u x2)

L7058

Hypothesis H22 : SNo (g x x2)

L7059

Hypothesis H23 : SNo (g x2 z + g y y2)

L7060

Hypothesis H24 : SNo (g x2 y2)

L7061

Theorem. (Conj_mul_SNo_assoc_lem2__74__16)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is fab960... and proposition id is 6bfd94...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__75__7

L7067

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__75__7 TMMorLf3eqGHEzELmJyJsoFgGtg87n5F76V bounty of about 25 bars ***)

L7068

Variable x : set

L7069

Variable y : set

L7070

Variable z : set

L7071

Variable w : set

L7072

Variable u : set

L7073

Variable v : set

L7074

Variable x2 : set

L7075

Variable y2 : set

L7076

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7077

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7078

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7079

Hypothesis H3 : SNo x

L7080

Hypothesis H4 : SNo y

L7081

Hypothesis H5 : SNo z

L7082

Hypothesis H6 : SNo (g x y)

L7083

Hypothesis H8 : u ∈ SNoL x

L7084

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7085

Hypothesis H10 : SNo u

L7086

Hypothesis H11 : u < x

L7087

Hypothesis H12 : SNo v

L7088

Hypothesis H13 : x2 ∈ SNoR y

L7089

Hypothesis H14 : y2 ∈ SNoL z

L7090

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7091

Hypothesis H16 : SNo x2

L7092

Hypothesis H17 : y < x2

L7093

Hypothesis H18 : SNo y2

L7094

Hypothesis H19 : y2 < z

L7095

Hypothesis H20 : SNo (g u y)

L7096

Hypothesis H21 : SNo (g u x2)

L7097

Hypothesis H22 : SNo (g x x2)

L7098

Hypothesis H23 : SNo (g x2 z + g y y2)

L7099

Theorem. (Conj_mul_SNo_assoc_lem2__75__7)

SNo (g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 6be245... and proposition id is a28af2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__75__19

L7105

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__75__19 TMchhuUvq6awkL2mN7LoeSWVi6FF4Y3SVg1 bounty of about 25 bars ***)

L7106

Variable x : set

L7107

Variable y : set

L7108

Variable z : set

L7109

Variable w : set

L7110

Variable u : set

L7111

Variable v : set

L7112

Variable x2 : set

L7113

Variable y2 : set

L7114

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7115

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7116

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7117

Hypothesis H3 : SNo x

L7118

Hypothesis H4 : SNo y

L7119

Hypothesis H5 : SNo z

L7120

Hypothesis H6 : SNo (g x y)

L7121

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7122

Hypothesis H8 : u ∈ SNoL x

L7123

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7124

Hypothesis H10 : SNo u

L7125

Hypothesis H11 : u < x

L7126

Hypothesis H12 : SNo v

L7127

Hypothesis H13 : x2 ∈ SNoR y

L7128

Hypothesis H14 : y2 ∈ SNoL z

L7129

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7130

Hypothesis H16 : SNo x2

L7131

Hypothesis H17 : y < x2

L7132

Hypothesis H18 : SNo y2

L7133

Hypothesis H20 : SNo (g u y)

L7134

Hypothesis H21 : SNo (g u x2)

L7135

Hypothesis H22 : SNo (g x x2)

L7136

Hypothesis H23 : SNo (g x2 z + g y y2)

L7137

Theorem. (Conj_mul_SNo_assoc_lem2__75__19)

SNo (g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 722dc5... and proposition id is 78894f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__77__18

L7143

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__77__18 TMUMAy59dAPD9HP7pRVgCbiDvr5HL8zYYtz bounty of about 25 bars ***)

L7144

Variable x : set

L7145

Variable y : set

L7146

Variable z : set

L7147

Variable w : set

L7148

Variable u : set

L7149

Variable v : set

L7150

Variable x2 : set

L7151

Variable y2 : set

L7152

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7153

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7154

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7155

Hypothesis H3 : SNo x

L7156

Hypothesis H4 : SNo y

L7157

Hypothesis H5 : SNo z

L7158

Hypothesis H6 : SNo (g x y)

L7159

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7160

Hypothesis H8 : u ∈ SNoL x

L7161

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7162

Hypothesis H10 : SNo u

L7163

Hypothesis H11 : u < x

L7164

Hypothesis H12 : SNo v

L7165

Hypothesis H13 : x2 ∈ SNoR y

L7166

Hypothesis H14 : y2 ∈ SNoL z

L7167

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7168

Hypothesis H16 : SNo x2

L7169

Hypothesis H17 : y < x2

L7170

Hypothesis H19 : y2 < z

L7171

Hypothesis H20 : SNo (g u y)

L7172

Hypothesis H21 : SNo (g u x2)

L7173

Hypothesis H22 : SNo (g x x2)

L7174

Hypothesis H23 : SNo (g x2 z)

L7175

Theorem. (Conj_mul_SNo_assoc_lem2__77__18)

SNo (g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is 8a9219... and proposition id is dd93fe...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__79__17

L7181

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__79__17 TMdRrSrTCeg1aBJ1SK9CAT3ffoWHUhNNRUa bounty of about 25 bars ***)

L7182

Variable x : set

L7183

Variable y : set

L7184

Variable z : set

L7185

Variable w : set

L7186

Variable u : set

L7187

Variable v : set

L7188

Variable x2 : set

L7189

Variable y2 : set

L7190

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7191

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7192

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7193

Hypothesis H3 : SNo x

L7194

Hypothesis H4 : SNo y

L7195

Hypothesis H5 : SNo z

L7196

Hypothesis H6 : SNo (g x y)

L7197

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7198

Hypothesis H8 : u ∈ SNoL x

L7199

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7200

Hypothesis H10 : SNo u

L7201

Hypothesis H11 : u < x

L7202

Hypothesis H12 : SNo v

L7203

Hypothesis H13 : x2 ∈ SNoR y

L7204

Hypothesis H14 : y2 ∈ SNoL z

L7205

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7206

Hypothesis H16 : SNo x2

L7207

Hypothesis H18 : SNo y2

L7208

Hypothesis H19 : y2 < z

L7209

Hypothesis H20 : SNo (g u y)

L7210

Hypothesis H21 : SNo (g u x2)

L7211

Theorem. (Conj_mul_SNo_assoc_lem2__79__17)

SNo (g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 0bd594... and proposition id is 1ac883...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__79__19

L7217

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__79__19 TMa4hApiuyiLbznCpqgq76oCUPSeaoupGSj bounty of about 25 bars ***)

L7218

Variable x : set

L7219

Variable y : set

L7220

Variable z : set

L7221

Variable w : set

L7222

Variable u : set

L7223

Variable v : set

L7224

Variable x2 : set

L7225

Variable y2 : set

L7226

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7227

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7228

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7229

Hypothesis H3 : SNo x

L7230

Hypothesis H4 : SNo y

L7231

Hypothesis H5 : SNo z

L7232

Hypothesis H6 : SNo (g x y)

L7233

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7234

Hypothesis H8 : u ∈ SNoL x

L7235

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7236

Hypothesis H10 : SNo u

L7237

Hypothesis H11 : u < x

L7238

Hypothesis H12 : SNo v

L7239

Hypothesis H13 : x2 ∈ SNoR y

L7240

Hypothesis H14 : y2 ∈ SNoL z

L7241

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7242

Hypothesis H16 : SNo x2

L7243

Hypothesis H17 : y < x2

L7244

Hypothesis H18 : SNo y2

L7245

Hypothesis H20 : SNo (g u y)

L7246

Hypothesis H21 : SNo (g u x2)

L7247

Theorem. (Conj_mul_SNo_assoc_lem2__79__19)

SNo (g x x2) → g (g x y) z < w

In Proofgold the corresponding term root is 158ac9... and proposition id is 2e74b9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__80__11

L7253

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__80__11 TMGxXVew6ifmMvWXt4SU9PvcPMzfFWArYri bounty of about 25 bars ***)

L7254

Variable x : set

L7255

Variable y : set

L7256

Variable z : set

L7257

Variable w : set

L7258

Variable u : set

L7259

Variable v : set

L7260

Variable x2 : set

L7261

Variable y2 : set

L7262

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7263

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7264

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7265

Hypothesis H3 : SNo x

L7266

Hypothesis H4 : SNo y

L7267

Hypothesis H5 : SNo z

L7268

Hypothesis H6 : SNo (g x y)

L7269

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7270

Hypothesis H8 : u ∈ SNoL x

L7271

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7272

Hypothesis H10 : SNo u

L7273

Hypothesis H12 : SNo v

L7274

Hypothesis H13 : x2 ∈ SNoR y

L7275

Hypothesis H14 : y2 ∈ SNoL z

L7276

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7277

Hypothesis H16 : SNo x2

L7278

Hypothesis H17 : y < x2

L7279

Hypothesis H18 : SNo y2

L7280

Hypothesis H19 : y2 < z

L7281

Hypothesis H20 : SNo (g u y)

L7282

Theorem. (Conj_mul_SNo_assoc_lem2__80__11)

SNo (g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is ab4424... and proposition id is 65849d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__81__19

L7288

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__81__19 TMTVcGt5Jks2ECokwC99x99ouDacQAZWhk8 bounty of about 25 bars ***)

L7289

Variable x : set

L7290

Variable y : set

L7291

Variable z : set

L7292

Variable w : set

L7293

Variable u : set

L7294

Variable v : set

L7295

Variable x2 : set

L7296

Variable y2 : set

L7297

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7298

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7299

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7300

Hypothesis H3 : SNo x

L7301

Hypothesis H4 : SNo y

L7302

Hypothesis H5 : SNo z

L7303

Hypothesis H6 : SNo (g x y)

L7304

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7305

Hypothesis H8 : u ∈ SNoL x

L7306

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7307

Hypothesis H10 : SNo u

L7308

Hypothesis H11 : u < x

L7309

Hypothesis H12 : SNo v

L7310

Hypothesis H13 : x2 ∈ SNoR y

L7311

Hypothesis H14 : y2 ∈ SNoL z

L7312

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7313

Hypothesis H16 : SNo x2

L7314

Hypothesis H17 : y < x2

L7315

Hypothesis H18 : SNo y2

L7316

Theorem. (Conj_mul_SNo_assoc_lem2__81__19)

SNo (g u y) → g (g x y) z < w

In Proofgold the corresponding term root is 522235... and proposition id is bd7f70...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__82__13

L7322

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__82__13 TMLTkDaEidxHX7pTvG74zaRz2pqUWdtuM99 bounty of about 25 bars ***)

L7323

Variable x : set

L7324

Variable y : set

L7325

Variable z : set

L7326

Variable w : set

L7327

Variable u : set

L7328

Variable v : set

L7329

Variable x2 : set

L7330

Variable y2 : set

L7331

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7332

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7333

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7334

Hypothesis H3 : SNo x

L7335

Hypothesis H4 : SNo y

L7336

Hypothesis H5 : SNo z

L7337

Hypothesis H6 : SNo (g x y)

L7338

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7339

Hypothesis H8 : u ∈ SNoL x

L7340

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7341

Hypothesis H10 : SNo u

L7342

Hypothesis H11 : u < x

L7343

Hypothesis H12 : SNo v

L7344

Hypothesis H14 : y2 ∈ SNoR z

L7345

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7346

Hypothesis H16 : SNo x2

L7347

Hypothesis H17 : x2 < y

L7348

Hypothesis H18 : SNo y2

L7349

Hypothesis H19 : z < y2

L7350

Hypothesis H20 : SNo (g x2 z + g y y2)

L7351

Hypothesis H21 : SNo (v + g x2 y2)

L7352

Hypothesis H22 : SNo (g x (v + g x2 y2))

L7353

Hypothesis H23 : SNo (g u (v + g x2 y2))

L7354

Hypothesis H24 : SNo (g u (g x2 z + g y y2))

L7355

Hypothesis H25 : SNo (g x (g x2 z + g y y2))

L7356

Hypothesis H26 : SNo (g u y + g x x2)

L7357

Hypothesis H27 : SNo (g (g x y + g u x2) z)

L7358

Hypothesis H28 : SNo (g (g u y + g x x2) z)

L7359

Hypothesis H29 : SNo (g (g x y + g u x2) y2)

L7360

Theorem. (Conj_mul_SNo_assoc_lem2__82__13)

SNo (g (g u y + g x x2) y2) → g (g x y) z < w

In Proofgold the corresponding term root is 192e91... and proposition id is 8bd810...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__83__29

L7366

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__83__29 TMW5TTAGgQ7cixwS3g8HL9pR5xo1RxRbE21 bounty of about 25 bars ***)

L7367

Variable x : set

L7368

Variable y : set

L7369

Variable z : set

L7370

Variable w : set

L7371

Variable u : set

L7372

Variable v : set

L7373

Variable x2 : set

L7374

Variable y2 : set

L7375

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7376

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7377

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7378

Hypothesis H3 : SNo x

L7379

Hypothesis H4 : SNo y

L7380

Hypothesis H5 : SNo z

L7381

Hypothesis H6 : SNo (g x y)

L7382

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7383

Hypothesis H8 : u ∈ SNoL x

L7384

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7385

Hypothesis H10 : SNo u

L7386

Hypothesis H11 : u < x

L7387

Hypothesis H12 : SNo v

L7388

Hypothesis H13 : x2 ∈ SNoL y

L7389

Hypothesis H14 : y2 ∈ SNoR z

L7390

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7391

Hypothesis H16 : SNo x2

L7392

Hypothesis H17 : x2 < y

L7393

Hypothesis H18 : SNo y2

L7394

Hypothesis H19 : z < y2

L7395

Hypothesis H20 : SNo (g x2 z + g y y2)

L7396

Hypothesis H21 : SNo (v + g x2 y2)

L7397

Hypothesis H22 : SNo (g x (v + g x2 y2))

L7398

Hypothesis H23 : SNo (g u (v + g x2 y2))

L7399

Hypothesis H24 : SNo (g u (g x2 z + g y y2))

L7400

Hypothesis H25 : SNo (g x (g x2 z + g y y2))

L7401

Hypothesis H26 : SNo (g u y + g x x2)

L7402

Hypothesis H27 : SNo (g (g x y + g u x2) z)

L7403

Hypothesis H28 : SNo (g (g u y + g x x2) z)

L7404

Theorem. (Conj_mul_SNo_assoc_lem2__83__29)

SNo (g (g x y + g u x2) y2) → g (g x y) z < w

In Proofgold the corresponding term root is 5e297f... and proposition id is fa777c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__84__12

L7410

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__84__12 TMFodmi4xxS7aGL6CfBbmtkPpeUhqgEjby8 bounty of about 25 bars ***)

L7411

Variable x : set

L7412

Variable y : set

L7413

Variable z : set

L7414

Variable w : set

L7415

Variable u : set

L7416

Variable v : set

L7417

Variable x2 : set

L7418

Variable y2 : set

L7419

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7420

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7421

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7422

Hypothesis H3 : SNo x

L7423

Hypothesis H4 : SNo y

L7424

Hypothesis H5 : SNo z

L7425

Hypothesis H6 : SNo (g x y)

L7426

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7427

Hypothesis H8 : u ∈ SNoL x

L7428

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7429

Hypothesis H10 : SNo u

L7430

Hypothesis H11 : u < x

L7431

Hypothesis H13 : x2 ∈ SNoL y

L7432

Hypothesis H14 : y2 ∈ SNoR z

L7433

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7434

Hypothesis H16 : SNo x2

L7435

Hypothesis H17 : x2 < y

L7436

Hypothesis H18 : SNo y2

L7437

Hypothesis H19 : z < y2

L7438

Hypothesis H20 : SNo (g x2 z + g y y2)

L7439

Hypothesis H21 : SNo (v + g x2 y2)

L7440

Hypothesis H22 : SNo (g x (v + g x2 y2))

L7441

Hypothesis H23 : SNo (g u (v + g x2 y2))

L7442

Hypothesis H24 : SNo (g u (g x2 z + g y y2))

L7443

Hypothesis H25 : SNo (g x (g x2 z + g y y2))

L7444

Hypothesis H26 : SNo (g u y + g x x2)

L7445

Hypothesis H27 : SNo (g (g x y + g u x2) z)

L7446

Hypothesis H28 : SNo (g (g u y + g x x2) z)

L7447

Theorem. (Conj_mul_SNo_assoc_lem2__84__12)

SNo (g x y + g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is 802687... and proposition id is 0e0e3a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__84__14

L7453

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__84__14 TMQY4T3ubJfhftBv9ViH96G4KYSEWsCieLH bounty of about 25 bars ***)

L7454

Variable x : set

L7455

Variable y : set

L7456

Variable z : set

L7457

Variable w : set

L7458

Variable u : set

L7459

Variable v : set

L7460

Variable x2 : set

L7461

Variable y2 : set

L7462

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7463

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7464

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7465

Hypothesis H3 : SNo x

L7466

Hypothesis H4 : SNo y

L7467

Hypothesis H5 : SNo z

L7468

Hypothesis H6 : SNo (g x y)

L7469

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7470

Hypothesis H8 : u ∈ SNoL x

L7471

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7472

Hypothesis H10 : SNo u

L7473

Hypothesis H11 : u < x

L7474

Hypothesis H12 : SNo v

L7475

Hypothesis H13 : x2 ∈ SNoL y

L7476

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7477

Hypothesis H16 : SNo x2

L7478

Hypothesis H17 : x2 < y

L7479

Hypothesis H18 : SNo y2

L7480

Hypothesis H19 : z < y2

L7481

Hypothesis H20 : SNo (g x2 z + g y y2)

L7482

Hypothesis H21 : SNo (v + g x2 y2)

L7483

Hypothesis H22 : SNo (g x (v + g x2 y2))

L7484

Hypothesis H23 : SNo (g u (v + g x2 y2))

L7485

Hypothesis H24 : SNo (g u (g x2 z + g y y2))

L7486

Hypothesis H25 : SNo (g x (g x2 z + g y y2))

L7487

Hypothesis H26 : SNo (g u y + g x x2)

L7488

Hypothesis H27 : SNo (g (g x y + g u x2) z)

L7489

Hypothesis H28 : SNo (g (g u y + g x x2) z)

L7490

Theorem. (Conj_mul_SNo_assoc_lem2__84__14)

SNo (g x y + g u x2) → g (g x y) z < w

In Proofgold the corresponding term root is 2e755c... and proposition id is 577b2f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__86__1

L7496

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__86__1 TMbtP3vCJ1MBrpjDbBZsnQyASGZqSeMueqC bounty of about 25 bars ***)

L7497

Variable x : set

L7498

Variable y : set

L7499

Variable z : set

L7500

Variable w : set

L7501

Variable u : set

L7502

Variable v : set

L7503

Variable x2 : set

L7504

Variable y2 : set

L7505

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7506

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7507

Hypothesis H3 : SNo x

L7508

Hypothesis H4 : SNo y

L7509

Hypothesis H5 : SNo z

L7510

Hypothesis H6 : SNo (g x y)

L7511

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7512

Hypothesis H8 : u ∈ SNoL x

L7513

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7514

Hypothesis H10 : SNo u

L7515

Hypothesis H11 : u < x

L7516

Hypothesis H12 : SNo v

L7517

Hypothesis H13 : x2 ∈ SNoL y

L7518

Hypothesis H14 : y2 ∈ SNoR z

L7519

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7520

Hypothesis H16 : SNo x2

L7521

Hypothesis H17 : x2 < y

L7522

Hypothesis H18 : SNo y2

L7523

Hypothesis H19 : z < y2

L7524

Hypothesis H20 : SNo (g x2 z + g y y2)

L7525

Hypothesis H21 : SNo (v + g x2 y2)

L7526

Hypothesis H22 : SNo (g x (v + g x2 y2))

L7527

Hypothesis H23 : SNo (g u (v + g x2 y2))

L7528

Hypothesis H24 : SNo (g u (g x2 z + g y y2))

L7529

Hypothesis H25 : SNo (g x (g x2 z + g y y2))

L7530

Hypothesis H26 : SNo (g x y + g u x2)

L7531

Hypothesis H27 : SNo (g u y + g x x2)

L7532

Theorem. (Conj_mul_SNo_assoc_lem2__86__1)

SNo (g (g x y + g u x2) z) → g (g x y) z < w

In Proofgold the corresponding term root is ca64eb... and proposition id is 2f6ad0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__86__26

L7538

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__86__26 TMcvDyNzPQM4UuPfpLtJ2gJnaYhUKLhWokY bounty of about 25 bars ***)

L7539

Variable x : set

L7540

Variable y : set

L7541

Variable z : set

L7542

Variable w : set

L7543

Variable u : set

L7544

Variable v : set

L7545

Variable x2 : set

L7546

Variable y2 : set

L7547

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7548

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7549

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7550

Hypothesis H3 : SNo x

L7551

Hypothesis H4 : SNo y

L7552

Hypothesis H5 : SNo z

L7553

Hypothesis H6 : SNo (g x y)

L7554

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7555

Hypothesis H8 : u ∈ SNoL x

L7556

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7557

Hypothesis H10 : SNo u

L7558

Hypothesis H11 : u < x

L7559

Hypothesis H12 : SNo v

L7560

Hypothesis H13 : x2 ∈ SNoL y

L7561

Hypothesis H14 : y2 ∈ SNoR z

L7562

Hypothesis H15 : (g x2 z + g y y2) ≤ v + g x2 y2

L7563

Hypothesis H16 : SNo x2

L7564

Hypothesis H17 : x2 < y

L7565

Hypothesis H18 : SNo y2

L7566

Hypothesis H19 : z < y2

L7567

Hypothesis H20 : SNo (g x2 z + g y y2)

L7568

Hypothesis H21 : SNo (v + g x2 y2)

L7569

Hypothesis H22 : SNo (g x (v + g x2 y2))

L7570

Hypothesis H23 : SNo (g u (v + g x2 y2))

L7571

Hypothesis H24 : SNo (g u (g x2 z + g y y2))

L7572

Hypothesis H25 : SNo (g x (g x2 z + g y y2))

L7573

Hypothesis H27 : SNo (g u y + g x x2)

L7574

Theorem. (Conj_mul_SNo_assoc_lem2__86__26)

SNo (g (g x y + g u x2) z) → g (g x y) z < w

In Proofgold the corresponding term root is e45887... and proposition id is 8a375a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__91__7

L7580

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__91__7 TMakxsvUDJMgbHQUTUbQeaem1Md2B2Xrfio bounty of about 25 bars ***)

L7581

Variable x : set

L7582

Variable y : set

L7583

Variable z : set

L7584

Variable w : set

L7585

Variable u : set

L7586

Variable v : set

L7587

Variable x2 : set

L7588

Variable y2 : set

L7589

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7590

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7591

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7592

Hypothesis H3 : SNo x

L7593

Hypothesis H4 : SNo y

L7594

Hypothesis H5 : SNo z

L7595

Hypothesis H6 : SNo (g x y)

L7596

Hypothesis H8 : u ∈ SNoL x

L7597

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7598

Hypothesis H10 : SNo u

L7599

Hypothesis H11 : u < x

L7600

Hypothesis H12 : SNo v

L7601

Hypothesis H13 : SNo (g u v)

L7602

Hypothesis H14 : SNo (g u y)

L7603

Hypothesis H15 : x2 ∈ SNoL y

L7604

Hypothesis H16 : y2 ∈ SNoR z

L7605

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7606

Hypothesis H18 : SNo x2

L7607

Hypothesis H19 : x2 < y

L7608

Hypothesis H20 : SNo y2

L7609

Hypothesis H21 : z < y2

L7610

Hypothesis H22 : SNo (g x x2)

L7611

Hypothesis H23 : SNo (g x2 z + g y y2)

L7612

Hypothesis H24 : SNo (v + g x2 y2)

L7613

Hypothesis H25 : SNo (g x (v + g x2 y2))

L7614

Hypothesis H26 : SNo (g u (v + g x2 y2))

L7615

Hypothesis H27 : SNo (g u (g x2 y2))

L7616

Theorem. (Conj_mul_SNo_assoc_lem2__91__7)

SNo (g u (g x2 z + g y y2)) → g (g x y) z < w

In Proofgold the corresponding term root is ab180f... and proposition id is def516...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__93__22

L7622

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__93__22 TMJaxFd1ija3gF4PD7gSpeAscbPBNgHF6Mz bounty of about 25 bars ***)

L7623

Variable x : set

L7624

Variable y : set

L7625

Variable z : set

L7626

Variable w : set

L7627

Variable u : set

L7628

Variable v : set

L7629

Variable x2 : set

L7630

Variable y2 : set

L7631

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7632

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7633

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7634

Hypothesis H3 : SNo x

L7635

Hypothesis H4 : SNo y

L7636

Hypothesis H5 : SNo z

L7637

Hypothesis H6 : SNo (g x y)

L7638

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7639

Hypothesis H8 : u ∈ SNoL x

L7640

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7641

Hypothesis H10 : SNo u

L7642

Hypothesis H11 : u < x

L7643

Hypothesis H12 : SNo v

L7644

Hypothesis H13 : SNo (g u v)

L7645

Hypothesis H14 : SNo (g u y)

L7646

Hypothesis H15 : x2 ∈ SNoL y

L7647

Hypothesis H16 : y2 ∈ SNoR z

L7648

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7649

Hypothesis H18 : SNo x2

L7650

Hypothesis H19 : x2 < y

L7651

Hypothesis H20 : SNo y2

L7652

Hypothesis H21 : z < y2

L7653

Hypothesis H23 : SNo (g x2 z + g y y2)

L7654

Hypothesis H24 : SNo (g x2 y2)

L7655

Hypothesis H25 : SNo (v + g x2 y2)

L7656

Hypothesis H26 : SNo (g x (v + g x2 y2))

L7657

Theorem. (Conj_mul_SNo_assoc_lem2__93__22)

SNo (g u (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is a34868... and proposition id is fd06a4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__94__7

L7663

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__94__7 TMHF55wZpTug6AbxH1Y8GFb88AfQ3itcx6N bounty of about 25 bars ***)

L7664

Variable x : set

L7665

Variable y : set

L7666

Variable z : set

L7667

Variable w : set

L7668

Variable u : set

L7669

Variable v : set

L7670

Variable x2 : set

L7671

Variable y2 : set

L7672

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7673

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7674

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7675

Hypothesis H3 : SNo x

L7676

Hypothesis H4 : SNo y

L7677

Hypothesis H5 : SNo z

L7678

Hypothesis H6 : SNo (g x y)

L7679

Hypothesis H8 : u ∈ SNoL x

L7680

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7681

Hypothesis H10 : SNo u

L7682

Hypothesis H11 : u < x

L7683

Hypothesis H12 : SNo v

L7684

Hypothesis H13 : SNo (g u v)

L7685

Hypothesis H14 : SNo (g u y)

L7686

Hypothesis H15 : x2 ∈ SNoL y

L7687

Hypothesis H16 : y2 ∈ SNoR z

L7688

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7689

Hypothesis H18 : SNo x2

L7690

Hypothesis H19 : x2 < y

L7691

Hypothesis H20 : SNo y2

L7692

Hypothesis H21 : z < y2

L7693

Hypothesis H22 : SNo (g x x2)

L7694

Hypothesis H23 : SNo (g x2 z + g y y2)

L7695

Hypothesis H24 : SNo (g x2 y2)

L7696

Hypothesis H25 : SNo (v + g x2 y2)

L7697

Theorem. (Conj_mul_SNo_assoc_lem2__94__7)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 22d745... and proposition id is ddd155...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__94__13

L7703

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__94__13 TMVnpiXyAyCcbtPj5bkfTyxRiQwHtjy9Fyv bounty of about 25 bars ***)

L7704

Variable x : set

L7705

Variable y : set

L7706

Variable z : set

L7707

Variable w : set

L7708

Variable u : set

L7709

Variable v : set

L7710

Variable x2 : set

L7711

Variable y2 : set

L7712

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7713

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7714

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7715

Hypothesis H3 : SNo x

L7716

Hypothesis H4 : SNo y

L7717

Hypothesis H5 : SNo z

L7718

Hypothesis H6 : SNo (g x y)

L7719

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7720

Hypothesis H8 : u ∈ SNoL x

L7721

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7722

Hypothesis H10 : SNo u

L7723

Hypothesis H11 : u < x

L7724

Hypothesis H12 : SNo v

L7725

Hypothesis H14 : SNo (g u y)

L7726

Hypothesis H15 : x2 ∈ SNoL y

L7727

Hypothesis H16 : y2 ∈ SNoR z

L7728

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7729

Hypothesis H18 : SNo x2

L7730

Hypothesis H19 : x2 < y

L7731

Hypothesis H20 : SNo y2

L7732

Hypothesis H21 : z < y2

L7733

Hypothesis H22 : SNo (g x x2)

L7734

Hypothesis H23 : SNo (g x2 z + g y y2)

L7735

Hypothesis H24 : SNo (g x2 y2)

L7736

Hypothesis H25 : SNo (v + g x2 y2)

L7737

Theorem. (Conj_mul_SNo_assoc_lem2__94__13)

SNo (g x (v + g x2 y2)) → g (g x y) z < w

In Proofgold the corresponding term root is 30b35c... and proposition id is 32ec6f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__95__2

L7743

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__95__2 TMUvij3kfTp9mS4idDBvEUhXmvZdBPHd6Fz bounty of about 25 bars ***)

L7744

Variable x : set

L7745

Variable y : set

L7746

Variable z : set

L7747

Variable w : set

L7748

Variable u : set

L7749

Variable v : set

L7750

Variable x2 : set

L7751

Variable y2 : set

L7752

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7753

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7754

Hypothesis H3 : SNo x

L7755

Hypothesis H4 : SNo y

L7756

Hypothesis H5 : SNo z

L7757

Hypothesis H6 : SNo (g x y)

L7758

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7759

Hypothesis H8 : u ∈ SNoL x

L7760

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7761

Hypothesis H10 : SNo u

L7762

Hypothesis H11 : u < x

L7763

Hypothesis H12 : SNo v

L7764

Hypothesis H13 : SNo (g u v)

L7765

Hypothesis H14 : SNo (g u y)

L7766

Hypothesis H15 : x2 ∈ SNoL y

L7767

Hypothesis H16 : y2 ∈ SNoR z

L7768

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7769

Hypothesis H18 : SNo x2

L7770

Hypothesis H19 : x2 < y

L7771

Hypothesis H20 : SNo y2

L7772

Hypothesis H21 : z < y2

L7773

Hypothesis H22 : SNo (g x x2)

L7774

Hypothesis H23 : SNo (g x2 z + g y y2)

L7775

Hypothesis H24 : SNo (g x2 y2)

L7776

Theorem. (Conj_mul_SNo_assoc_lem2__95__2)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is a808a5... and proposition id is 7bc14a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__95__8

L7782

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__95__8 TMRYeijPBdhiw71G2ubNa29GehSKppYYzZQ bounty of about 25 bars ***)

L7783

Variable x : set

L7784

Variable y : set

L7785

Variable z : set

L7786

Variable w : set

L7787

Variable u : set

L7788

Variable v : set

L7789

Variable x2 : set

L7790

Variable y2 : set

L7791

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7792

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7793

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7794

Hypothesis H3 : SNo x

L7795

Hypothesis H4 : SNo y

L7796

Hypothesis H5 : SNo z

L7797

Hypothesis H6 : SNo (g x y)

L7798

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7799

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7800

Hypothesis H10 : SNo u

L7801

Hypothesis H11 : u < x

L7802

Hypothesis H12 : SNo v

L7803

Hypothesis H13 : SNo (g u v)

L7804

Hypothesis H14 : SNo (g u y)

L7805

Hypothesis H15 : x2 ∈ SNoL y

L7806

Hypothesis H16 : y2 ∈ SNoR z

L7807

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7808

Hypothesis H18 : SNo x2

L7809

Hypothesis H19 : x2 < y

L7810

Hypothesis H20 : SNo y2

L7811

Hypothesis H21 : z < y2

L7812

Hypothesis H22 : SNo (g x x2)

L7813

Hypothesis H23 : SNo (g x2 z + g y y2)

L7814

Hypothesis H24 : SNo (g x2 y2)

L7815

Theorem. (Conj_mul_SNo_assoc_lem2__95__8)

SNo (v + g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is b4765c... and proposition id is bc2efd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__96__20

L7821

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__96__20 TMJxri3fAstdKePrGbv4qLDKFHLrWXhKXZJ bounty of about 25 bars ***)

L7822

Variable x : set

L7823

Variable y : set

L7824

Variable z : set

L7825

Variable w : set

L7826

Variable u : set

L7827

Variable v : set

L7828

Variable x2 : set

L7829

Variable y2 : set

L7830

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7831

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7832

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7833

Hypothesis H3 : SNo x

L7834

Hypothesis H4 : SNo y

L7835

Hypothesis H5 : SNo z

L7836

Hypothesis H6 : SNo (g x y)

L7837

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7838

Hypothesis H8 : u ∈ SNoL x

L7839

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7840

Hypothesis H10 : SNo u

L7841

Hypothesis H11 : u < x

L7842

Hypothesis H12 : SNo v

L7843

Hypothesis H13 : SNo (g u v)

L7844

Hypothesis H14 : SNo (g u y)

L7845

Hypothesis H15 : x2 ∈ SNoL y

L7846

Hypothesis H16 : y2 ∈ SNoR z

L7847

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7848

Hypothesis H18 : SNo x2

L7849

Hypothesis H19 : x2 < y

L7850

Hypothesis H21 : z < y2

L7851

Hypothesis H22 : SNo (g x x2)

L7852

Hypothesis H23 : SNo (g x2 z + g y y2)

L7853

Theorem. (Conj_mul_SNo_assoc_lem2__96__20)

SNo (g x2 y2) → g (g x y) z < w

In Proofgold the corresponding term root is 3c37aa... and proposition id is ab0107...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__97__24

L7859

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__97__24 TMN9nXAUDGdWiUJ5xNUzJW1oDfrPSx2zC9q bounty of about 25 bars ***)

L7860

Variable x : set

L7861

Variable y : set

L7862

Variable z : set

L7863

Variable w : set

L7864

Variable u : set

L7865

Variable v : set

L7866

Variable x2 : set

L7867

Variable y2 : set

L7868

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7869

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7870

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7871

Hypothesis H3 : SNo x

L7872

Hypothesis H4 : SNo y

L7873

Hypothesis H5 : SNo z

L7874

Hypothesis H6 : SNo (g x y)

L7875

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7876

Hypothesis H8 : u ∈ SNoL x

L7877

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7878

Hypothesis H10 : SNo u

L7879

Hypothesis H11 : u < x

L7880

Hypothesis H12 : SNo v

L7881

Hypothesis H13 : SNo (g u v)

L7882

Hypothesis H14 : SNo (g u y)

L7883

Hypothesis H15 : x2 ∈ SNoL y

L7884

Hypothesis H16 : y2 ∈ SNoR z

L7885

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7886

Hypothesis H18 : SNo x2

L7887

Hypothesis H19 : x2 < y

L7888

Hypothesis H20 : SNo y2

L7889

Hypothesis H21 : z < y2

L7890

Hypothesis H22 : SNo (g x x2)

L7891

Hypothesis H23 : SNo (g x2 z)

L7892

Theorem. (Conj_mul_SNo_assoc_lem2__97__24)

SNo (g x2 z + g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is cf60e8... and proposition id is b87f9a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__98__8

L7898

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__98__8 TMKeoxuV4Rw6QfdQd1E5KuoKx9QtFnJNfEW bounty of about 25 bars ***)

L7899

Variable x : set

L7900

Variable y : set

L7901

Variable z : set

L7902

Variable w : set

L7903

Variable u : set

L7904

Variable v : set

L7905

Variable x2 : set

L7906

Variable y2 : set

L7907

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7908

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7909

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7910

Hypothesis H3 : SNo x

L7911

Hypothesis H4 : SNo y

L7912

Hypothesis H5 : SNo z

L7913

Hypothesis H6 : SNo (g x y)

L7914

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7915

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7916

Hypothesis H10 : SNo u

L7917

Hypothesis H11 : u < x

L7918

Hypothesis H12 : SNo v

L7919

Hypothesis H13 : SNo (g u v)

L7920

Hypothesis H14 : SNo (g u y)

L7921

Hypothesis H15 : x2 ∈ SNoL y

L7922

Hypothesis H16 : y2 ∈ SNoR z

L7923

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7924

Hypothesis H18 : SNo x2

L7925

Hypothesis H19 : x2 < y

L7926

Hypothesis H20 : SNo y2

L7927

Hypothesis H21 : z < y2

L7928

Hypothesis H22 : SNo (g x x2)

L7929

Hypothesis H23 : SNo (g x2 z)

L7930

Theorem. (Conj_mul_SNo_assoc_lem2__98__8)

SNo (g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is a2070b... and proposition id is 77a309...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__98__11

L7936

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__98__11 TMKbVDX8fgnRgqHnjiv1ustdKyctyGNDj87 bounty of about 25 bars ***)

L7937

Variable x : set

L7938

Variable y : set

L7939

Variable z : set

L7940

Variable w : set

L7941

Variable u : set

L7942

Variable v : set

L7943

Variable x2 : set

L7944

Variable y2 : set

L7945

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7946

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7947

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7948

Hypothesis H3 : SNo x

L7949

Hypothesis H4 : SNo y

L7950

Hypothesis H5 : SNo z

L7951

Hypothesis H6 : SNo (g x y)

L7952

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7953

Hypothesis H8 : u ∈ SNoL x

L7954

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7955

Hypothesis H10 : SNo u

L7956

Hypothesis H12 : SNo v

L7957

Hypothesis H13 : SNo (g u v)

L7958

Hypothesis H14 : SNo (g u y)

L7959

Hypothesis H15 : x2 ∈ SNoL y

L7960

Hypothesis H16 : y2 ∈ SNoR z

L7961

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L7962

Hypothesis H18 : SNo x2

L7963

Hypothesis H19 : x2 < y

L7964

Hypothesis H20 : SNo y2

L7965

Hypothesis H21 : z < y2

L7966

Hypothesis H22 : SNo (g x x2)

L7967

Hypothesis H23 : SNo (g x2 z)

L7968

Theorem. (Conj_mul_SNo_assoc_lem2__98__11)

SNo (g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is 8fd032... and proposition id is 545eb3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__98__16

L7974

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__98__16 TMJU3C4xLSttVHjJ6ncmvAzf7BSuNbPXEuB bounty of about 25 bars ***)

L7975

Variable x : set

L7976

Variable y : set

L7977

Variable z : set

L7978

Variable w : set

L7979

Variable u : set

L7980

Variable v : set

L7981

Variable x2 : set

L7982

Variable y2 : set

L7983

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L7984

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L7985

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L7986

Hypothesis H3 : SNo x

L7987

Hypothesis H4 : SNo y

L7988

Hypothesis H5 : SNo z

L7989

Hypothesis H6 : SNo (g x y)

L7990

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L7991

Hypothesis H8 : u ∈ SNoL x

L7992

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L7993

Hypothesis H10 : SNo u

L7994

Hypothesis H11 : u < x

L7995

Hypothesis H12 : SNo v

L7996

Hypothesis H13 : SNo (g u v)

L7997

Hypothesis H14 : SNo (g u y)

L7998

Hypothesis H15 : x2 ∈ SNoL y

L7999

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L8000

Hypothesis H18 : SNo x2

L8001

Hypothesis H19 : x2 < y

L8002

Hypothesis H20 : SNo y2

L8003

Hypothesis H21 : z < y2

L8004

Hypothesis H22 : SNo (g x x2)

L8005

Hypothesis H23 : SNo (g x2 z)

L8006

Theorem. (Conj_mul_SNo_assoc_lem2__98__16)

SNo (g y y2) → g (g x y) z < w

In Proofgold the corresponding term root is 265f16... and proposition id is 1c0e2d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__99__3

L8012

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__99__3 TMXv9o9WGT7KPrmUVv24TQB9qKGu8VWoJih bounty of about 25 bars ***)

L8013

Variable x : set

L8014

Variable y : set

L8015

Variable z : set

L8016

Variable w : set

L8017

Variable u : set

L8018

Variable v : set

L8019

Variable x2 : set

L8020

Variable y2 : set

L8021

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L8022

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L8023

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L8024

Hypothesis H4 : SNo y

L8025

Hypothesis H5 : SNo z

L8026

Hypothesis H6 : SNo (g x y)

L8027

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L8028

Hypothesis H8 : u ∈ SNoL x

L8029

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L8030

Hypothesis H10 : SNo u

L8031

Hypothesis H11 : u < x

L8032

Hypothesis H12 : SNo v

L8033

Hypothesis H13 : SNo (g u v)

L8034

Hypothesis H14 : SNo (g u y)

L8035

Hypothesis H15 : x2 ∈ SNoL y

L8036

Hypothesis H16 : y2 ∈ SNoR z

L8037

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L8038

Hypothesis H18 : SNo x2

L8039

Hypothesis H19 : x2 < y

L8040

Hypothesis H20 : SNo y2

L8041

Hypothesis H21 : z < y2

L8042

Hypothesis H22 : SNo (g x x2)

L8043

Theorem. (Conj_mul_SNo_assoc_lem2__99__3)

SNo (g x2 z) → g (g x y) z < w

In Proofgold the corresponding term root is 0bbb4b... and proposition id is 2a10c1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__99__8

L8049

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__99__8 TMHKWPxWcaPS7jnCyrj6dsZAFFm4JYkuHPr bounty of about 25 bars ***)

L8050

Variable x : set

L8051

Variable y : set

L8052

Variable z : set

L8053

Variable w : set

L8054

Variable u : set

L8055

Variable v : set

L8056

Variable x2 : set

L8057

Variable y2 : set

L8058

Hypothesis H0 : (∀z2 : set, ∀w2 : set, SNo z2 → SNo w2 → SNo (g z2 w2))

L8059

Hypothesis H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 < z2 → v2 < w2 → (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2)

L8060

Hypothesis H2 : (∀z2 : set, ∀w2 : set, ∀u2 : set, ∀v2 : set, SNo z2 → SNo w2 → SNo u2 → SNo v2 → u2 ≤ z2 → v2 ≤ w2 → (g u2 w2 + g z2 v2) ≤ g z2 w2 + g u2 v2)

L8061

Hypothesis H3 : SNo x

L8062

Hypothesis H4 : SNo y

L8063

Hypothesis H5 : SNo z

L8064

Hypothesis H6 : SNo (g x y)

L8065

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, SNo w2 → (∀u2 : set, u2 ∈ SNoS_ (SNoLev y) → (∀v2 : set, v2 ∈ SNoS_ (SNoLev z) → w = g z2 (g y z) + g x w2 + - (g z2 w2) → (g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) ≤ g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) → (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 → g (g x y) z < w))))

L8066

Hypothesis H9 : w = g u (g y z) + g x v + - (g u v)

L8067

Hypothesis H10 : SNo u

L8068

Hypothesis H11 : u < x

L8069

Hypothesis H12 : SNo v

L8070

Hypothesis H13 : SNo (g u v)

L8071

Hypothesis H14 : SNo (g u y)

L8072

Hypothesis H15 : x2 ∈ SNoL y

L8073

Hypothesis H16 : y2 ∈ SNoR z

L8074

Hypothesis H17 : (g x2 z + g y y2) ≤ v + g x2 y2

L8075

Hypothesis H18 : SNo x2

L8076

Hypothesis H19 : x2 < y

L8077

Hypothesis H20 : SNo y2

L8078

Hypothesis H21 : z < y2

L8079

Hypothesis H22 : SNo (g x x2)

L8080

Theorem. (Conj_mul_SNo_assoc_lem2__99__8)

SNo (g x2 z) → g (g x y) z < w

In Proofgold the corresponding term root is 948072... and proposition id is 7b303e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem2__101__4

L8086

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem2__101__4 TMc2SsF2ShVXAbGTuHuu7yxMqnUS24H358Y bounty of about 25 bars ***)

L8087

Variable x : set

L8088

Variable y : set

L8089

Variable z : set

L8090

Variable w : set

L8091

Variable u : set

L8092

Variable v : set

L8093

Hypothesis H0 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → SNo (g x2 y2))

L8094

Hypothesis H1 : (∀x2 : set, ∀y2 : set, SNo x2 → SNo y2 → (∀z2 : set, z2 ∈ SNoR (g x2 y2) → (∀P : prop, (∀w2 : set, w2 ∈ SNoL x2 → (∀u2 : set, u2 ∈ SNoR y2 → (g w2 y2 + g x2 u2) ≤ z2 + g w2 u2 → P)) → (∀w2 : set, w2 ∈ SNoR x2 → (∀u2 : set, u2 ∈ SNoL y2 → (g w2 y2 + g x2 u2) ≤ z2 + g w2 u2 → P)) → P)))

L8095

Hypothesis H2 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 < x2 → w2 < y2 → (g z2 y2 + g x2 w2) < g x2 y2 + g z2 w2)

L8096

Hypothesis H3 : (∀x2 : set, ∀y2 : set, ∀z2 : set, ∀w2 : set, SNo x2 → SNo y2 → SNo z2 → SNo w2 → z2 ≤ x2 → w2 ≤ y2 → (g z2 y2 + g x2 w2) ≤ g x2 y2 + g z2 w2)

L8097

Hypothesis H5 : SNo y

L8098

Hypothesis H6 : SNo z

L8099

Hypothesis H7 : SNo (g x y)

L8100

Hypothesis H8 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, SNo y2 → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → w = g x2 (g y z) + g x y2 + - (g x2 y2) → (g x (g z2 z + g y w2) + g x2 (y2 + g z2 w2)) ≤ g x2 (g z2 z + g y w2) + g x (y2 + g z2 w2) → (g (g x y + g x2 z2) z + g (g x2 y + g x z2) w2) < g (g x2 y + g x z2) z + g (g x y + g x2 z2) w2 → g (g x y) z < w))))

L8101

Hypothesis H9 : u ∈ SNoL x

L8102

Hypothesis H10 : v ∈ SNoR (g y z)

L8103

Hypothesis H11 : w = g u (g y z) + g x v + - (g u v)

L8104

Hypothesis H12 : SNo u

L8105

Hypothesis H13 : u < x

L8106

Hypothesis H14 : SNo v

L8107

Hypothesis H15 : SNo (g u v)

L8108

Theorem. (Conj_mul_SNo_assoc_lem2__101__4)

SNo (g u y) → g (g x y) z < w

In Proofgold the corresponding term root is 493f78... and proposition id is d2bfd1...

Proof:
Proof not loaded.