Beginning of Section Conj_mul_SNo_Lt__23__10

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

Variable x : set

(*** Conj_mul_SNo_Lt__23__10 TMSJzQCLFKAzZP39iMF3SeiYqfkQEMGroHm bounty of about 25 bars ***)

Variable y : set

Variable z : set

Variable w : set

Variable u : set

Hypothesis H0 : SNo x

L10

Hypothesis H1 : SNo z

L11

Hypothesis H2 : SNo w

L12

Hypothesis H3 : SNo (x * y)

L13

Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L14

Hypothesis H5 : SNo (z * y)

L15

Hypothesis H6 : SNo (x * w)

L16

Hypothesis H7 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L17

Hypothesis H8 : SNo (z * w)

L18

Hypothesis H9 : SNo (z * y + x * w)

L19

Hypothesis H11 : z ∈ SNoL x

L20

Hypothesis H12 : SNo u

L21

Hypothesis H13 : w < u

L22

Hypothesis H14 : SNoLev u ∈ SNoLev w

L23

Hypothesis H15 : u ∈ SNoL y

L24

Theorem. (Conj_mul_SNo_Lt__23__10)

u ∈ SNoR w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 470ad9... and proposition id is 63f307...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__24__6

L30

Variable x : set

(*** Conj_mul_SNo_Lt__24__6 TMM5B3jeWUhr9Y2AMqqvCKXtF7M5rCSGXGB bounty of about 25 bars ***)

L31

Variable y : set

L32

Variable z : set

L33

Variable w : set

L34

Variable u : set

L35

Hypothesis H0 : SNo x

L36

Hypothesis H1 : SNo y

L37

Hypothesis H2 : SNo z

L38

Hypothesis H3 : SNo w

L39

Hypothesis H4 : SNo (x * y)

L40

Hypothesis H5 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L41

Hypothesis H7 : SNo (x * w)

L42

Hypothesis H8 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L43

Hypothesis H9 : SNo (z * w)

L44

Hypothesis H10 : SNo (z * y + x * w)

L45

Hypothesis H11 : SNo (x * y + z * w)

L46

Hypothesis H12 : z ∈ SNoL x

L47

Hypothesis H13 : SNo u

L48

Hypothesis H14 : w < u

L49

Hypothesis H15 : u < y

L50

Hypothesis H16 : SNoLev u ∈ SNoLev w

L51

Hypothesis H17 : SNoLev u ∈ SNoLev y

L52

Theorem. (Conj_mul_SNo_Lt__24__6)

u ∈ SNoL y → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 2e623a... and proposition id is 2bfa44...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__24__9

L58

Variable x : set

(*** Conj_mul_SNo_Lt__24__9 TMR6L3XCy7nHCdGqXLydM6J2cu2xvTcaEvE bounty of about 25 bars ***)

L59

Variable y : set

L60

Variable z : set

L61

Variable w : set

L62

Variable u : set

L63

Hypothesis H0 : SNo x

L64

Hypothesis H1 : SNo y

L65

Hypothesis H2 : SNo z

L66

Hypothesis H3 : SNo w

L67

Hypothesis H4 : SNo (x * y)

L68

Hypothesis H5 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L69

Hypothesis H6 : SNo (z * y)

L70

Hypothesis H7 : SNo (x * w)

L71

Hypothesis H8 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L72

Hypothesis H10 : SNo (z * y + x * w)

L73

Hypothesis H11 : SNo (x * y + z * w)

L74

Hypothesis H12 : z ∈ SNoL x

L75

Hypothesis H13 : SNo u

L76

Hypothesis H14 : w < u

L77

Hypothesis H15 : u < y

L78

Hypothesis H16 : SNoLev u ∈ SNoLev w

L79

Hypothesis H17 : SNoLev u ∈ SNoLev y

L80

Theorem. (Conj_mul_SNo_Lt__24__9)

u ∈ SNoL y → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 7a2d12... and proposition id is 341bad...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__26__4

L86

Variable x : set

(*** Conj_mul_SNo_Lt__26__4 TMSh2aLEUxhrMRD3zK47Wk536jTYnEgofS4 bounty of about 25 bars ***)

L87

Variable y : set

L88

Variable z : set

L89

Variable w : set

L90

Variable u : set

L91

Hypothesis H0 : SNo (x * y)

L92

Hypothesis H1 : SNo (z * y)

L93

Hypothesis H2 : SNo (x * w)

L94

Hypothesis H3 : SNo (z * w)

L95

Hypothesis H5 : SNo (z * y + x * w)

L96

Hypothesis H6 : SNo (x * y + z * w)

L97

Hypothesis H7 : u ∈ SNoR z

L98

Hypothesis H8 : SNo (u * y)

L99

Hypothesis H9 : SNo (u * w)

L100

Hypothesis H10 : SNo (z * w + u * y)

L101

Hypothesis H11 : SNo (u * w + x * y)

L102

Hypothesis H12 : SNo (x * w + u * y)

L103

Hypothesis H13 : SNo (u * w + z * y)

L104

Hypothesis H14 : y ∈ SNoR w

L105

Hypothesis H15 : (x * w + u * y) < u * w + x * y

L106

Theorem. (Conj_mul_SNo_Lt__26__4)

(u * w + z * y) < z * w + u * y → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 8b64f4... and proposition id is c49d57...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__27__13

L112

Variable x : set

(*** Conj_mul_SNo_Lt__27__13 TMWDCi488kgJBFgbimVVes2Si3XcVHjwmyV bounty of about 25 bars ***)

L113

Variable y : set

L114

Variable z : set

L115

Variable w : set

L116

Variable u : set

L117

Hypothesis H0 : SNo (x * y)

L118

Hypothesis H1 : SNo (z * y)

L119

Hypothesis H2 : SNo (x * w)

L120

Hypothesis H3 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L121

Hypothesis H4 : SNo (z * w)

L122

Hypothesis H5 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L123

Hypothesis H6 : SNo (z * y + x * w)

L124

Hypothesis H7 : SNo (x * y + z * w)

L125

Hypothesis H8 : u ∈ SNoL x

L126

Hypothesis H9 : u ∈ SNoR z

L127

Hypothesis H10 : SNo (u * y)

L128

Hypothesis H11 : SNo (u * w)

L129

Hypothesis H12 : SNo (z * w + u * y)

L130

Hypothesis H14 : SNo (x * w + u * y)

L131

Hypothesis H15 : SNo (u * w + z * y)

L132

Hypothesis H16 : y ∈ SNoR w

L133

Theorem. (Conj_mul_SNo_Lt__27__13)

(x * w + u * y) < u * w + x * y → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is c6344f... and proposition id is 2688d7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__29__0

L139

Variable x : set

(*** Conj_mul_SNo_Lt__29__0 TMTA4dcGiWCpsNH7JKXsHZSxv7KQxCw912S bounty of about 25 bars ***)

L140

Variable y : set

L141

Variable z : set

L142

Variable w : set

L143

Variable u : set

L144

Hypothesis H1 : SNo (z * y)

L145

Hypothesis H2 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L146

Hypothesis H3 : SNo (x * w)

L147

Hypothesis H4 : SNo (z * w)

L148

Hypothesis H5 : SNo (z * y + x * w)

L149

Hypothesis H6 : SNo (x * y + z * w)

L150

Hypothesis H7 : u ∈ SNoR z

L151

Hypothesis H8 : SNo (u * y)

L152

Hypothesis H9 : SNo (u * w)

L153

Hypothesis H10 : SNo (u * y + x * w)

L154

Hypothesis H11 : SNo (u * y + z * w)

L155

Hypothesis H12 : SNo (x * y + u * w)

L156

Hypothesis H13 : SNo (z * y + u * w)

L157

Hypothesis H14 : w ∈ SNoL y

L158

Hypothesis H15 : (u * y + x * w) < x * y + u * w

L159

Theorem. (Conj_mul_SNo_Lt__29__0)

(z * y + u * w) < u * y + z * w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 613dc4... and proposition id is 47c069...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__29__6

L165

Variable x : set

(*** Conj_mul_SNo_Lt__29__6 TMWhvRjMoLy1WfwXqsJnxWVp2rUh7U6UWtY bounty of about 25 bars ***)

L166

Variable y : set

L167

Variable z : set

L168

Variable w : set

L169

Variable u : set

L170

Hypothesis H0 : SNo (x * y)

L171

Hypothesis H1 : SNo (z * y)

L172

Hypothesis H2 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L173

Hypothesis H3 : SNo (x * w)

L174

Hypothesis H4 : SNo (z * w)

L175

Hypothesis H5 : SNo (z * y + x * w)

L176

Hypothesis H7 : u ∈ SNoR z

L177

Hypothesis H8 : SNo (u * y)

L178

Hypothesis H9 : SNo (u * w)

L179

Hypothesis H10 : SNo (u * y + x * w)

L180

Hypothesis H11 : SNo (u * y + z * w)

L181

Hypothesis H12 : SNo (x * y + u * w)

L182

Hypothesis H13 : SNo (z * y + u * w)

L183

Hypothesis H14 : w ∈ SNoL y

L184

Hypothesis H15 : (u * y + x * w) < x * y + u * w

L185

Theorem. (Conj_mul_SNo_Lt__29__6)

(z * y + u * w) < u * y + z * w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 29b223... and proposition id is ee4bd8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__29__10

L191

Variable x : set

(*** Conj_mul_SNo_Lt__29__10 TMKCfb9ckE56MyrzjUBFqG6zkWFq4k7tXKE bounty of about 25 bars ***)

L192

Variable y : set

L193

Variable z : set

L194

Variable w : set

L195

Variable u : set

L196

Hypothesis H0 : SNo (x * y)

L197

Hypothesis H1 : SNo (z * y)

L198

Hypothesis H2 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L199

Hypothesis H3 : SNo (x * w)

L200

Hypothesis H4 : SNo (z * w)

L201

Hypothesis H5 : SNo (z * y + x * w)

L202

Hypothesis H6 : SNo (x * y + z * w)

L203

Hypothesis H7 : u ∈ SNoR z

L204

Hypothesis H8 : SNo (u * y)

L205

Hypothesis H9 : SNo (u * w)

L206

Hypothesis H11 : SNo (u * y + z * w)

L207

Hypothesis H12 : SNo (x * y + u * w)

L208

Hypothesis H13 : SNo (z * y + u * w)

L209

Hypothesis H14 : w ∈ SNoL y

L210

Hypothesis H15 : (u * y + x * w) < x * y + u * w

L211

Theorem. (Conj_mul_SNo_Lt__29__10)

(z * y + u * w) < u * y + z * w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is ce4345... and proposition id is 82e9e4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__31__9

L217

Variable x : set

(*** Conj_mul_SNo_Lt__31__9 TMH5Grq7zsnjYv26ZDa8JowokvgfSzbBM6g bounty of about 25 bars ***)

L218

Variable y : set

L219

Variable z : set

L220

Variable w : set

L221

Variable u : set

L222

Hypothesis H0 : SNo y

L223

Hypothesis H1 : SNo w

L224

Hypothesis H2 : w < y

L225

Hypothesis H3 : SNo (x * y)

L226

Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L227

Hypothesis H5 : SNo (z * y)

L228

Hypothesis H6 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L229

Hypothesis H7 : SNo (x * w)

L230

Hypothesis H8 : SNo (z * w)

L231

Hypothesis H10 : SNo (x * y + z * w)

L232

Hypothesis H11 : u ∈ SNoL x

L233

Hypothesis H12 : u ∈ SNoR z

L234

Hypothesis H13 : SNo (u * y)

L235

Hypothesis H14 : SNo (u * w)

L236

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

L237

Hypothesis H16 : SNo (u * y + z * w)

L238

Hypothesis H17 : SNo (x * y + u * w)

L239

Hypothesis H18 : SNo (z * y + u * w)

L240

Hypothesis H19 : SNoLev w ∈ SNoLev y

L241

Theorem. (Conj_mul_SNo_Lt__31__9)

w ∈ SNoL y → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 55d6c9... and proposition id is b8cac6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__32__13

L247

Variable x : set

(*** Conj_mul_SNo_Lt__32__13 TML5bFhRXz3dvDHaDREG312xhDsUeajvaPE bounty of about 25 bars ***)

L248

Variable y : set

L249

Variable z : set

L250

Variable w : set

L251

Variable u : set

L252

Variable v : set

L253

Hypothesis H0 : SNo (x * y)

L254

Hypothesis H1 : SNo (z * y)

L255

Hypothesis H2 : SNo (x * w)

L256

Hypothesis H3 : SNo (z * w)

L257

Hypothesis H4 : SNo (z * y + x * w)

L258

Hypothesis H5 : SNo (x * y + z * w)

L259

Hypothesis H6 : SNo (u * y)

L260

Hypothesis H7 : SNo (u * w)

L261

Hypothesis H8 : SNo (x * v)

L262

Hypothesis H9 : SNo (z * v)

L263

Hypothesis H10 : SNo (u * v)

L264

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L265

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L266

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L267

Hypothesis H15 : SNo (u * v + u * v)

L268

Hypothesis H16 : SNo (u * y + z * v)

L269

Hypothesis H17 : SNo (u * w + x * v)

L270

Hypothesis H18 : SNo (z * y + u * v)

L271

Hypothesis H19 : SNo (x * w + u * v)

L272

Hypothesis H20 : SNo (u * w + z * v)

L273

Hypothesis H21 : SNo (u * y + x * v)

L274

Hypothesis H22 : SNo (x * y + u * v)

L275

Theorem. (Conj_mul_SNo_Lt__32__13)

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

In Proofgold the corresponding term root is d71cf3... and proposition id is 060be0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__33__2

L281

Variable x : set

(*** Conj_mul_SNo_Lt__33__2 TMHSmdzHtuLUHZGW6ztqu8vyfVuQq5WVEDR bounty of about 25 bars ***)

L282

Variable y : set

L283

Variable z : set

L284

Variable w : set

L285

Variable u : set

L286

Variable v : set

L287

Hypothesis H0 : SNo (x * y)

L288

Hypothesis H1 : SNo (z * y)

L289

Hypothesis H3 : SNo (z * w)

L290

Hypothesis H4 : SNo (z * y + x * w)

L291

Hypothesis H5 : SNo (x * y + z * w)

L292

Hypothesis H6 : SNo (u * y)

L293

Hypothesis H7 : SNo (u * w)

L294

Hypothesis H8 : SNo (x * v)

L295

Hypothesis H9 : SNo (z * v)

L296

Hypothesis H10 : SNo (u * v)

L297

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L298

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L299

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L300

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L301

Hypothesis H15 : SNo (u * v + u * v)

L302

Hypothesis H16 : SNo (u * y + z * v)

L303

Hypothesis H17 : SNo (u * w + x * v)

L304

Hypothesis H18 : SNo (z * y + u * v)

L305

Hypothesis H19 : SNo (x * w + u * v)

L306

Hypothesis H20 : SNo (u * w + z * v)

L307

Hypothesis H21 : SNo (u * y + x * v)

L308

Theorem. (Conj_mul_SNo_Lt__33__2)

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

In Proofgold the corresponding term root is 455322... and proposition id is ab71e3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__33__21

L314

Variable x : set

(*** Conj_mul_SNo_Lt__33__21 TMYzh6zx9rAk9HCGP926gPZqUq8g7rmcKhC bounty of about 25 bars ***)

L315

Variable y : set

L316

Variable z : set

L317

Variable w : set

L318

Variable u : set

L319

Variable v : set

L320

Hypothesis H0 : SNo (x * y)

L321

Hypothesis H1 : SNo (z * y)

L322

Hypothesis H2 : SNo (x * w)

L323

Hypothesis H3 : SNo (z * w)

L324

Hypothesis H4 : SNo (z * y + x * w)

L325

Hypothesis H5 : SNo (x * y + z * w)

L326

Hypothesis H6 : SNo (u * y)

L327

Hypothesis H7 : SNo (u * w)

L328

Hypothesis H8 : SNo (x * v)

L329

Hypothesis H9 : SNo (z * v)

L330

Hypothesis H10 : SNo (u * v)

L331

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L332

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L333

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L334

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L335

Hypothesis H15 : SNo (u * v + u * v)

L336

Hypothesis H16 : SNo (u * y + z * v)

L337

Hypothesis H17 : SNo (u * w + x * v)

L338

Hypothesis H18 : SNo (z * y + u * v)

L339

Hypothesis H19 : SNo (x * w + u * v)

L340

Hypothesis H20 : SNo (u * w + z * v)

L341

Theorem. (Conj_mul_SNo_Lt__33__21)

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

In Proofgold the corresponding term root is 7bd88f... and proposition id is a62a05...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__34__7

L347

Variable x : set

(*** Conj_mul_SNo_Lt__34__7 TMYJt5rndtZh7Rk1pKGYha7EXJ7yzy66wt3 bounty of about 25 bars ***)

L348

Variable y : set

L349

Variable z : set

L350

Variable w : set

L351

Variable u : set

L352

Variable v : set

L353

Hypothesis H0 : SNo (x * y)

L354

Hypothesis H1 : SNo (z * y)

L355

Hypothesis H2 : SNo (x * w)

L356

Hypothesis H3 : SNo (z * w)

L357

Hypothesis H4 : SNo (z * y + x * w)

L358

Hypothesis H5 : SNo (x * y + z * w)

L359

Hypothesis H6 : SNo (u * y)

L360

Hypothesis H8 : SNo (x * v)

L361

Hypothesis H9 : SNo (z * v)

L362

Hypothesis H10 : SNo (u * v)

L363

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L364

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L365

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L366

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L367

Hypothesis H15 : SNo (u * v + u * v)

L368

Hypothesis H16 : SNo (u * y + z * v)

L369

Hypothesis H17 : SNo (u * w + x * v)

L370

Hypothesis H18 : SNo (z * y + u * v)

L371

Hypothesis H19 : SNo (x * w + u * v)

L372

Hypothesis H20 : SNo (u * w + z * v)

L373

Theorem. (Conj_mul_SNo_Lt__34__7)

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

In Proofgold the corresponding term root is 2bbeee... and proposition id is 9ada80...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__35__7

L379

Variable x : set

(*** Conj_mul_SNo_Lt__35__7 TMJX18vHwEUsfnF9mNsDK9rbPzvoCZWqtu3 bounty of about 25 bars ***)

L380

Variable y : set

L381

Variable z : set

L382

Variable w : set

L383

Variable u : set

L384

Variable v : set

L385

Hypothesis H0 : SNo (x * y)

L386

Hypothesis H1 : SNo (z * y)

L387

Hypothesis H2 : SNo (x * w)

L388

Hypothesis H3 : SNo (z * w)

L389

Hypothesis H4 : SNo (z * y + x * w)

L390

Hypothesis H5 : SNo (x * y + z * w)

L391

Hypothesis H6 : SNo (u * y)

L392

Hypothesis H8 : SNo (x * v)

L393

Hypothesis H9 : SNo (z * v)

L394

Hypothesis H10 : SNo (u * v)

L395

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L396

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L397

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L398

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L399

Hypothesis H15 : SNo (u * v + u * v)

L400

Hypothesis H16 : SNo (u * y + z * v)

L401

Hypothesis H17 : SNo (u * w + x * v)

L402

Hypothesis H18 : SNo (z * y + u * v)

L403

Hypothesis H19 : SNo (x * w + u * v)

L404

Hypothesis H20 : SNo (u * w + z * v)

L405

Theorem. (Conj_mul_SNo_Lt__35__7)

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

In Proofgold the corresponding term root is ea7056... and proposition id is fe94c2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__35__18

L411

Variable x : set

(*** Conj_mul_SNo_Lt__35__18 TMK8wovVNU9kS2PZdgF6BhTCbUZPzycbTzX bounty of about 25 bars ***)

L412

Variable y : set

L413

Variable z : set

L414

Variable w : set

L415

Variable u : set

L416

Variable v : set

L417

Hypothesis H0 : SNo (x * y)

L418

Hypothesis H1 : SNo (z * y)

L419

Hypothesis H2 : SNo (x * w)

L420

Hypothesis H3 : SNo (z * w)

L421

Hypothesis H4 : SNo (z * y + x * w)

L422

Hypothesis H5 : SNo (x * y + z * w)

L423

Hypothesis H6 : SNo (u * y)

L424

Hypothesis H7 : SNo (u * w)

L425

Hypothesis H8 : SNo (x * v)

L426

Hypothesis H9 : SNo (z * v)

L427

Hypothesis H10 : SNo (u * v)

L428

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L429

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L430

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L431

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L432

Hypothesis H15 : SNo (u * v + u * v)

L433

Hypothesis H16 : SNo (u * y + z * v)

L434

Hypothesis H17 : SNo (u * w + x * v)

L435

Hypothesis H19 : SNo (x * w + u * v)

L436

Hypothesis H20 : SNo (u * w + z * v)

L437

Theorem. (Conj_mul_SNo_Lt__35__18)

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

In Proofgold the corresponding term root is de8939... and proposition id is 1ec8d1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__36__19

L443

Variable x : set

(*** Conj_mul_SNo_Lt__36__19 TMPGm6jKX5nLrSVYy17sf1nwYupMDqETvxB bounty of about 25 bars ***)

L444

Variable y : set

L445

Variable z : set

L446

Variable w : set

L447

Variable u : set

L448

Variable v : set

L449

Hypothesis H0 : SNo (x * y)

L450

Hypothesis H1 : SNo (z * y)

L451

Hypothesis H2 : SNo (x * w)

L452

Hypothesis H3 : SNo (z * w)

L453

Hypothesis H4 : SNo (z * y + x * w)

L454

Hypothesis H5 : SNo (x * y + z * w)

L455

Hypothesis H6 : SNo (u * y)

L456

Hypothesis H7 : SNo (u * w)

L457

Hypothesis H8 : SNo (x * v)

L458

Hypothesis H9 : SNo (z * v)

L459

Hypothesis H10 : SNo (u * v)

L460

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L461

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L462

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L463

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L464

Hypothesis H15 : SNo (u * v + u * v)

L465

Hypothesis H16 : SNo (u * y + z * v)

L466

Hypothesis H17 : SNo (u * w + x * v)

L467

Hypothesis H18 : SNo (z * y + u * v)

L468

Theorem. (Conj_mul_SNo_Lt__36__19)

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

In Proofgold the corresponding term root is d2a442... and proposition id is ec348c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__37__3

L474

Variable x : set

(*** Conj_mul_SNo_Lt__37__3 TMdd9s7BtiTMUjxLWLTJp2846TJHzXRToHF bounty of about 25 bars ***)

L475

Variable y : set

L476

Variable z : set

L477

Variable w : set

L478

Variable u : set

L479

Variable v : set

L480

Hypothesis H0 : SNo (x * y)

L481

Hypothesis H1 : SNo (z * y)

L482

Hypothesis H2 : SNo (x * w)

L483

Hypothesis H4 : SNo (z * y + x * w)

L484

Hypothesis H5 : SNo (x * y + z * w)

L485

Hypothesis H6 : SNo (u * y)

L486

Hypothesis H7 : SNo (u * w)

L487

Hypothesis H8 : SNo (x * v)

L488

Hypothesis H9 : SNo (z * v)

L489

Hypothesis H10 : SNo (u * v)

L490

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L491

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L492

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L493

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L494

Hypothesis H15 : SNo (u * v + u * v)

L495

Hypothesis H16 : SNo (u * y + z * v)

L496

Hypothesis H17 : SNo (u * w + x * v)

L497

Hypothesis H18 : SNo (z * y + u * v)

L498

Theorem. (Conj_mul_SNo_Lt__37__3)

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

In Proofgold the corresponding term root is c81e74... and proposition id is 43eb45...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__37__17

L504

Variable x : set

(*** Conj_mul_SNo_Lt__37__17 TMKxcFTVDGQoSDERxzXeqPSauXeZHy9CvHp bounty of about 25 bars ***)

L505

Variable y : set

L506

Variable z : set

L507

Variable w : set

L508

Variable u : set

L509

Variable v : set

L510

Hypothesis H0 : SNo (x * y)

L511

Hypothesis H1 : SNo (z * y)

L512

Hypothesis H2 : SNo (x * w)

L513

Hypothesis H3 : SNo (z * w)

L514

Hypothesis H4 : SNo (z * y + x * w)

L515

Hypothesis H5 : SNo (x * y + z * w)

L516

Hypothesis H6 : SNo (u * y)

L517

Hypothesis H7 : SNo (u * w)

L518

Hypothesis H8 : SNo (x * v)

L519

Hypothesis H9 : SNo (z * v)

L520

Hypothesis H10 : SNo (u * v)

L521

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L522

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L523

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L524

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L525

Hypothesis H15 : SNo (u * v + u * v)

L526

Hypothesis H16 : SNo (u * y + z * v)

L527

Hypothesis H18 : SNo (z * y + u * v)

L528

Theorem. (Conj_mul_SNo_Lt__37__17)

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

In Proofgold the corresponding term root is 92329e... and proposition id is 89b243...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__38__15

L534

Variable x : set

(*** Conj_mul_SNo_Lt__38__15 TMTwLsdnjqoLgpRJp2v1VacStBVAPrEGyy3 bounty of about 25 bars ***)

L535

Variable y : set

L536

Variable z : set

L537

Variable w : set

L538

Variable u : set

L539

Variable v : set

L540

Hypothesis H0 : SNo (x * y)

L541

Hypothesis H1 : SNo (z * y)

L542

Hypothesis H2 : SNo (x * w)

L543

Hypothesis H3 : SNo (z * w)

L544

Hypothesis H4 : SNo (z * y + x * w)

L545

Hypothesis H5 : SNo (x * y + z * w)

L546

Hypothesis H6 : SNo (u * y)

L547

Hypothesis H7 : SNo (u * w)

L548

Hypothesis H8 : SNo (x * v)

L549

Hypothesis H9 : SNo (z * v)

L550

Hypothesis H10 : SNo (u * v)

L551

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L552

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L553

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L554

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L555

Hypothesis H16 : SNo (u * y + z * v)

L556

Hypothesis H17 : SNo (u * w + x * v)

L557

Theorem. (Conj_mul_SNo_Lt__38__15)

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

In Proofgold the corresponding term root is 2237d1... and proposition id is ddb245...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__39__12

L563

Variable x : set

(*** Conj_mul_SNo_Lt__39__12 TMctgCSrCEZVM5vFET34N5JSrUnmWBYgtew bounty of about 25 bars ***)

L564

Variable y : set

L565

Variable z : set

L566

Variable w : set

L567

Variable u : set

L568

Variable v : set

L569

Hypothesis H0 : SNo (x * y)

L570

Hypothesis H1 : SNo (z * y)

L571

Hypothesis H2 : SNo (x * w)

L572

Hypothesis H3 : SNo (z * w)

L573

Hypothesis H4 : SNo (z * y + x * w)

L574

Hypothesis H5 : SNo (x * y + z * w)

L575

Hypothesis H6 : SNo (u * y)

L576

Hypothesis H7 : SNo (u * w)

L577

Hypothesis H8 : SNo (x * v)

L578

Hypothesis H9 : SNo (z * v)

L579

Hypothesis H10 : SNo (u * v)

L580

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L581

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L582

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L583

Hypothesis H15 : SNo (u * v + u * v)

L584

Hypothesis H16 : SNo (u * y + z * v)

L585

Theorem. (Conj_mul_SNo_Lt__39__12)

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

In Proofgold the corresponding term root is 63dabc... and proposition id is 9e443f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__40__13

L591

Variable x : set

(*** Conj_mul_SNo_Lt__40__13 TMbqwNM7jKX6BcQkodHxxQjRfX5NfsgQ5yG bounty of about 25 bars ***)

L592

Variable y : set

L593

Variable z : set

L594

Variable w : set

L595

Variable u : set

L596

Variable v : set

L597

Hypothesis H0 : SNo (x * y)

L598

Hypothesis H1 : SNo (z * y)

L599

Hypothesis H2 : SNo (x * w)

L600

Hypothesis H3 : SNo (z * w)

L601

Hypothesis H4 : SNo (z * y + x * w)

L602

Hypothesis H5 : SNo (x * y + z * w)

L603

Hypothesis H6 : SNo (u * y)

L604

Hypothesis H7 : SNo (u * w)

L605

Hypothesis H8 : SNo (x * v)

L606

Hypothesis H9 : SNo (z * v)

L607

Hypothesis H10 : SNo (u * v)

L608

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L609

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L610

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L611

Hypothesis H15 : SNo (u * v + u * v)

L612

Theorem. (Conj_mul_SNo_Lt__40__13)

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

In Proofgold the corresponding term root is ba378f... and proposition id is 09451a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__41__1

L618

Variable x : set

(*** Conj_mul_SNo_Lt__41__1 TMKm2eAorisw6JeUukkBJNer9GGzwAtMP5G bounty of about 25 bars ***)

L619

Variable y : set

L620

Variable z : set

L621

Variable w : set

L622

Variable u : set

L623

Variable v : set

L624

Hypothesis H0 : SNo (x * y)

L625

Hypothesis H2 : SNo (x * w)

L626

Hypothesis H3 : SNo (z * w)

L627

Hypothesis H4 : SNo (z * y + x * w)

L628

Hypothesis H5 : SNo (x * y + z * w)

L629

Hypothesis H6 : SNo (u * y)

L630

Hypothesis H7 : SNo (u * w)

L631

Hypothesis H8 : SNo (x * v)

L632

Hypothesis H9 : SNo (z * v)

L633

Hypothesis H10 : SNo (u * v)

L634

Hypothesis H11 : (u * y + x * v) < x * y + u * v

L635

Hypothesis H12 : (u * w + z * v) < z * w + u * v

L636

Hypothesis H13 : (x * w + u * v) < u * w + x * v

L637

Hypothesis H14 : (z * y + u * v) < u * y + z * v

L638

Theorem. (Conj_mul_SNo_Lt__41__1)

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

In Proofgold the corresponding term root is cc2d9e... and proposition id is a5eba2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__42__12

L644

Variable x : set

(*** Conj_mul_SNo_Lt__42__12 TMaV8cE235TovDCH811xBw5neyptMNXQMFH bounty of about 25 bars ***)

L645

Variable y : set

L646

Variable z : set

L647

Variable w : set

L648

Variable u : set

L649

Variable v : set

L650

Hypothesis H0 : SNo (x * y)

L651

Hypothesis H1 : SNo (z * y)

L652

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L653

Hypothesis H3 : SNo (x * w)

L654

Hypothesis H4 : SNo (z * w)

L655

Hypothesis H5 : SNo (z * y + x * w)

L656

Hypothesis H6 : SNo (x * y + z * w)

L657

Hypothesis H7 : u ∈ SNoR z

L658

Hypothesis H8 : SNo (u * y)

L659

Hypothesis H9 : SNo (u * w)

L660

Hypothesis H10 : v ∈ SNoL y

L661

Hypothesis H11 : SNo (x * v)

L662

Hypothesis H13 : SNo (u * v)

L663

Hypothesis H14 : (u * y + x * v) < x * y + u * v

L664

Hypothesis H15 : (u * w + z * v) < z * w + u * v

L665

Hypothesis H16 : (x * w + u * v) < u * w + x * v

L666

Theorem. (Conj_mul_SNo_Lt__42__12)

(z * y + u * v) < u * y + z * v → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is d78767... and proposition id is 657cc6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__44__1

L672

Variable x : set

(*** Conj_mul_SNo_Lt__44__1 TMarWgHWHQkTmPy73tiKhaP9gxa2Ko2J5oq bounty of about 25 bars ***)

L673

Variable y : set

L674

Variable z : set

L675

Variable w : set

L676

Variable u : set

L677

Variable v : set

L678

Hypothesis H0 : SNo (x * y)

L679

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L680

Hypothesis H3 : SNo (x * w)

L681

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L682

Hypothesis H5 : SNo (z * w)

L683

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L684

Hypothesis H7 : SNo (z * y + x * w)

L685

Hypothesis H8 : SNo (x * y + z * w)

L686

Hypothesis H9 : u ∈ SNoL x

L687

Hypothesis H10 : u ∈ SNoR z

L688

Hypothesis H11 : SNo (u * y)

L689

Hypothesis H12 : SNo (u * w)

L690

Hypothesis H13 : v ∈ SNoL y

L691

Hypothesis H14 : v ∈ SNoR w

L692

Hypothesis H15 : SNo (x * v)

L693

Hypothesis H16 : SNo (z * v)

L694

Hypothesis H17 : SNo (u * v)

L695

Hypothesis H18 : (u * y + x * v) < x * y + u * v

L696

Theorem. (Conj_mul_SNo_Lt__44__1)

(u * w + z * v) < z * w + u * v → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 0d724e... and proposition id is f5c363...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__44__18

L702

Variable x : set

(*** Conj_mul_SNo_Lt__44__18 TMNmBfTK1kGpyEDdYvSXzy4b6eRqVLTpsCB bounty of about 25 bars ***)

L703

Variable y : set

L704

Variable z : set

L705

Variable w : set

L706

Variable u : set

L707

Variable v : set

L708

Hypothesis H0 : SNo (x * y)

L709

Hypothesis H1 : SNo (z * y)

L710

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L711

Hypothesis H3 : SNo (x * w)

L712

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L713

Hypothesis H5 : SNo (z * w)

L714

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L715

Hypothesis H7 : SNo (z * y + x * w)

L716

Hypothesis H8 : SNo (x * y + z * w)

L717

Hypothesis H9 : u ∈ SNoL x

L718

Hypothesis H10 : u ∈ SNoR z

L719

Hypothesis H11 : SNo (u * y)

L720

Hypothesis H12 : SNo (u * w)

L721

Hypothesis H13 : v ∈ SNoL y

L722

Hypothesis H14 : v ∈ SNoR w

L723

Hypothesis H15 : SNo (x * v)

L724

Hypothesis H16 : SNo (z * v)

L725

Hypothesis H17 : SNo (u * v)

L726

Theorem. (Conj_mul_SNo_Lt__44__18)

(u * w + z * v) < z * w + u * v → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 4f77de... and proposition id is 5bbd27...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__48__2

L732

Variable x : set

(*** Conj_mul_SNo_Lt__48__2 TMWcibTE18PVDvGppPKzfoSnT4jy1GXTFUh bounty of about 25 bars ***)

L733

Variable y : set

L734

Variable z : set

L735

Variable w : set

L736

Variable u : set

L737

Variable v : set

L738

Hypothesis H0 : SNo x

L739

Hypothesis H1 : SNo z

L740

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → (x2 * y + x * y2) < x * y + x2 * y2))

L741

Hypothesis H4 : SNo (z * y)

L742

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L743

Hypothesis H6 : SNo (x * w)

L744

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L745

Hypothesis H8 : SNo (z * w)

L746

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L747

Hypothesis H10 : SNo (z * y + x * w)

L748

Hypothesis H11 : SNo (x * y + z * w)

L749

Hypothesis H12 : SNo u

L750

Hypothesis H13 : u ∈ SNoL x

L751

Hypothesis H14 : u ∈ SNoR z

L752

Hypothesis H15 : SNo (u * y)

L753

Hypothesis H16 : SNo (u * w)

L754

Hypothesis H17 : SNo v

L755

Hypothesis H18 : v ∈ SNoL y

L756

Hypothesis H19 : v ∈ SNoR w

L757

Theorem. (Conj_mul_SNo_Lt__48__2)

SNo (x * v) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 8ee58f... and proposition id is 313339...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__49__1

L763

Variable x : set

(*** Conj_mul_SNo_Lt__49__1 TML7tpJgU7asx354ehoVs16fvzJQRox2c4p bounty of about 25 bars ***)

L764

Variable y : set

L765

Variable z : set

L766

Variable w : set

L767

Variable u : set

L768

Variable v : set

L769

Hypothesis H0 : SNo x

L770

Hypothesis H2 : SNo w

L771

Hypothesis H3 : SNo (x * y)

L772

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → (x2 * y + x * y2) < x * y + x2 * y2))

L773

Hypothesis H5 : SNo (z * y)

L774

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L775

Hypothesis H7 : SNo (x * w)

L776

Hypothesis H8 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L777

Hypothesis H9 : SNo (z * w)

L778

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L779

Hypothesis H11 : SNo (z * y + x * w)

L780

Hypothesis H12 : SNo (x * y + z * w)

L781

Hypothesis H13 : SNo u

L782

Hypothesis H14 : u ∈ SNoL x

L783

Hypothesis H15 : u ∈ SNoR z

L784

Hypothesis H16 : SNo (u * y)

L785

Hypothesis H17 : SNo (u * w)

L786

Hypothesis H18 : SNo v

L787

Hypothesis H19 : w < v

L788

Hypothesis H20 : SNoLev v ∈ SNoLev w

L789

Hypothesis H21 : v ∈ SNoL y

L790

Theorem. (Conj_mul_SNo_Lt__49__1)

v ∈ SNoR w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 9ad680... and proposition id is 65ec26...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__49__12

L796

Variable x : set

(*** Conj_mul_SNo_Lt__49__12 TMMcV8MoX6xsdnu5dKdZ1TjjTet849b8Lgv bounty of about 25 bars ***)

L797

Variable y : set

L798

Variable z : set

L799

Variable w : set

L800

Variable u : set

L801

Variable v : set

L802

Hypothesis H0 : SNo x

L803

Hypothesis H1 : SNo z

L804

Hypothesis H2 : SNo w

L805

Hypothesis H3 : SNo (x * y)

L806

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → (x2 * y + x * y2) < x * y + x2 * y2))

L807

Hypothesis H5 : SNo (z * y)

L808

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L809

Hypothesis H7 : SNo (x * w)

L810

Hypothesis H8 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L811

Hypothesis H9 : SNo (z * w)

L812

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L813

Hypothesis H11 : SNo (z * y + x * w)

L814

Hypothesis H13 : SNo u

L815

Hypothesis H14 : u ∈ SNoL x

L816

Hypothesis H15 : u ∈ SNoR z

L817

Hypothesis H16 : SNo (u * y)

L818

Hypothesis H17 : SNo (u * w)

L819

Hypothesis H18 : SNo v

L820

Hypothesis H19 : w < v

L821

Hypothesis H20 : SNoLev v ∈ SNoLev w

L822

Hypothesis H21 : v ∈ SNoL y

L823

Theorem. (Conj_mul_SNo_Lt__49__12)

v ∈ SNoR w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 961d01... and proposition id is fd1b61...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__49__20

L829

Variable x : set

(*** Conj_mul_SNo_Lt__49__20 TMWXERb8eMLqk8KUUcmNcXrFHE9UKRtaEeS bounty of about 25 bars ***)

L830

Variable y : set

L831

Variable z : set

L832

Variable w : set

L833

Variable u : set

L834

Variable v : set

L835

Hypothesis H0 : SNo x

L836

Hypothesis H1 : SNo z

L837

Hypothesis H2 : SNo w

L838

Hypothesis H3 : SNo (x * y)

L839

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → (x2 * y + x * y2) < x * y + x2 * y2))

L840

Hypothesis H5 : SNo (z * y)

L841

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L842

Hypothesis H7 : SNo (x * w)

L843

Hypothesis H8 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L844

Hypothesis H9 : SNo (z * w)

L845

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L846

Hypothesis H11 : SNo (z * y + x * w)

L847

Hypothesis H12 : SNo (x * y + z * w)

L848

Hypothesis H13 : SNo u

L849

Hypothesis H14 : u ∈ SNoL x

L850

Hypothesis H15 : u ∈ SNoR z

L851

Hypothesis H16 : SNo (u * y)

L852

Hypothesis H17 : SNo (u * w)

L853

Hypothesis H18 : SNo v

L854

Hypothesis H19 : w < v

L855

Hypothesis H21 : v ∈ SNoL y

L856

Theorem. (Conj_mul_SNo_Lt__49__20)

v ∈ SNoR w → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 5c12cd... and proposition id is 2f413b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__50__16

L862

Variable x : set

(*** Conj_mul_SNo_Lt__50__16 TMH9BPEh4SpP41578uqDdQa1YkUvb4kREoT bounty of about 25 bars ***)

L863

Variable y : set

L864

Variable z : set

L865

Variable w : set

L866

Variable u : set

L867

Variable v : set

L868

Hypothesis H0 : SNo x

L869

Hypothesis H1 : SNo y

L870

Hypothesis H2 : SNo z

L871

Hypothesis H3 : SNo w

L872

Hypothesis H4 : SNo (x * y)

L873

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → (x2 * y + x * y2) < x * y + x2 * y2))

L874

Hypothesis H6 : SNo (z * y)

L875

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoL y → (z * y + x2 * y2) < x2 * y + z * y2))

L876

Hypothesis H8 : SNo (x * w)

L877

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR w → (x * w + x2 * y2) < x2 * w + x * y2))

L878

Hypothesis H10 : SNo (z * w)

L879

Hypothesis H11 : (∀x2 : set, x2 ∈ SNoR z → (∀y2 : set, y2 ∈ SNoR w → (x2 * w + z * y2) < z * w + x2 * y2))

L880

Hypothesis H12 : SNo (z * y + x * w)

L881

Hypothesis H13 : SNo (x * y + z * w)

L882

Hypothesis H14 : SNo u

L883

Hypothesis H15 : u ∈ SNoL x

L884

Hypothesis H17 : SNo (u * y)

L885

Hypothesis H18 : SNo (u * w)

L886

Hypothesis H19 : SNo v

L887

Hypothesis H20 : w < v

L888

Hypothesis H21 : v < y

L889

Hypothesis H22 : SNoLev v ∈ SNoLev w

L890

Hypothesis H23 : SNoLev v ∈ SNoLev y

L891

Theorem. (Conj_mul_SNo_Lt__50__16)

v ∈ SNoL y → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is c33c01... and proposition id is f63d4e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__52__23

L897

Variable x : set

(*** Conj_mul_SNo_Lt__52__23 TMKr37tuCDts2pzu2unAyMSL81ffCtuaTXh bounty of about 25 bars ***)

L898

Variable y : set

L899

Variable z : set

L900

Variable w : set

L901

Variable u : set

L902

Hypothesis H0 : SNo x

L903

Hypothesis H1 : SNo y

L904

Hypothesis H2 : SNo z

L905

Hypothesis H3 : SNo w

L906

Hypothesis H4 : w < y

L907

Hypothesis H5 : SNo (x * y)

L908

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L909

Hypothesis H7 : SNo (z * y)

L910

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L911

Hypothesis H9 : SNo (x * w)

L912

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L913

Hypothesis H11 : SNo (z * w)

L914

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L915

Hypothesis H13 : SNo (z * y + x * w)

L916

Hypothesis H14 : SNo (x * y + z * w)

L917

Hypothesis H15 : SNo u

L918

Hypothesis H16 : u ∈ SNoL x

L919

Hypothesis H17 : u ∈ SNoR z

L920

Hypothesis H18 : SNo (u * y)

L921

Hypothesis H19 : SNo (u * w)

L922

Hypothesis H20 : SNo (u * y + x * w)

L923

Hypothesis H21 : SNo (u * y + z * w)

L924

Hypothesis H22 : SNo (x * y + u * w)

L925

Hypothesis H24 : SNo (z * w + u * y)

L926

Hypothesis H25 : SNo (u * w + x * y)

L927

Theorem. (Conj_mul_SNo_Lt__52__23)

SNo (x * w + u * y) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is ba2a42... and proposition id is 136d67...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__53__23

L933

Variable x : set

(*** Conj_mul_SNo_Lt__53__23 TMSJjwr8ysXu9yGDAUBsonpGRxKVocKuQLs bounty of about 25 bars ***)

L934

Variable y : set

L935

Variable z : set

L936

Variable w : set

L937

Variable u : set

L938

Hypothesis H0 : SNo x

L939

Hypothesis H1 : SNo y

L940

Hypothesis H2 : SNo z

L941

Hypothesis H3 : SNo w

L942

Hypothesis H4 : w < y

L943

Hypothesis H5 : SNo (x * y)

L944

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L945

Hypothesis H7 : SNo (z * y)

L946

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L947

Hypothesis H9 : SNo (x * w)

L948

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L949

Hypothesis H11 : SNo (z * w)

L950

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L951

Hypothesis H13 : SNo (z * y + x * w)

L952

Hypothesis H14 : SNo (x * y + z * w)

L953

Hypothesis H15 : SNo u

L954

Hypothesis H16 : u ∈ SNoL x

L955

Hypothesis H17 : u ∈ SNoR z

L956

Hypothesis H18 : SNo (u * y)

L957

Hypothesis H19 : SNo (u * w)

L958

Hypothesis H20 : SNo (u * y + x * w)

L959

Hypothesis H21 : SNo (u * y + z * w)

L960

Hypothesis H22 : SNo (x * y + u * w)

L961

Hypothesis H24 : SNo (z * w + u * y)

L962

Theorem. (Conj_mul_SNo_Lt__53__23)

SNo (u * w + x * y) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is d387dc... and proposition id is b9d05d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__53__24

L968

Variable x : set

(*** Conj_mul_SNo_Lt__53__24 TMFPWZbUopHc3TdQ1J2Bes4zduTDY1dy6VZ bounty of about 25 bars ***)

L969

Variable y : set

L970

Variable z : set

L971

Variable w : set

L972

Variable u : set

L973

Hypothesis H0 : SNo x

L974

Hypothesis H1 : SNo y

L975

Hypothesis H2 : SNo z

L976

Hypothesis H3 : SNo w

L977

Hypothesis H4 : w < y

L978

Hypothesis H5 : SNo (x * y)

L979

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L980

Hypothesis H7 : SNo (z * y)

L981

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L982

Hypothesis H9 : SNo (x * w)

L983

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L984

Hypothesis H11 : SNo (z * w)

L985

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L986

Hypothesis H13 : SNo (z * y + x * w)

L987

Hypothesis H14 : SNo (x * y + z * w)

L988

Hypothesis H15 : SNo u

L989

Hypothesis H16 : u ∈ SNoL x

L990

Hypothesis H17 : u ∈ SNoR z

L991

Hypothesis H18 : SNo (u * y)

L992

Hypothesis H19 : SNo (u * w)

L993

Hypothesis H20 : SNo (u * y + x * w)

L994

Hypothesis H21 : SNo (u * y + z * w)

L995

Hypothesis H22 : SNo (x * y + u * w)

L996

Hypothesis H23 : SNo (z * y + u * w)

L997

Theorem. (Conj_mul_SNo_Lt__53__24)

SNo (u * w + x * y) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is b1f40a... and proposition id is 34c999...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__54__4

L1003

Variable x : set

(*** Conj_mul_SNo_Lt__54__4 TMWq1y2gqkj9p9nmSYM5VHHb35BLt2KU4xj bounty of about 25 bars ***)

L1004

Variable y : set

L1005

Variable z : set

L1006

Variable w : set

L1007

Variable u : set

L1008

Hypothesis H0 : SNo x

L1009

Hypothesis H1 : SNo y

L1010

Hypothesis H2 : SNo z

L1011

Hypothesis H3 : SNo w

L1012

Hypothesis H5 : SNo (x * y)

L1013

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1014

Hypothesis H7 : SNo (z * y)

L1015

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1016

Hypothesis H9 : SNo (x * w)

L1017

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1018

Hypothesis H11 : SNo (z * w)

L1019

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1020

Hypothesis H13 : SNo (z * y + x * w)

L1021

Hypothesis H14 : SNo (x * y + z * w)

L1022

Hypothesis H15 : SNo u

L1023

Hypothesis H16 : u ∈ SNoL x

L1024

Hypothesis H17 : u ∈ SNoR z

L1025

Hypothesis H18 : SNo (u * y)

L1026

Hypothesis H19 : SNo (u * w)

L1027

Hypothesis H20 : SNo (u * y + x * w)

L1028

Hypothesis H21 : SNo (u * y + z * w)

L1029

Hypothesis H22 : SNo (x * y + u * w)

L1030

Hypothesis H23 : SNo (z * y + u * w)

L1031

Theorem. (Conj_mul_SNo_Lt__54__4)

SNo (z * w + u * y) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 0c4dc0... and proposition id is 997f7f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__54__22

L1037

Variable x : set

(*** Conj_mul_SNo_Lt__54__22 TMbr6WaR47XVgTBQE5PtkpyBBQHX7tY2GRZ bounty of about 25 bars ***)

L1038

Variable y : set

L1039

Variable z : set

L1040

Variable w : set

L1041

Variable u : set

L1042

Hypothesis H0 : SNo x

L1043

Hypothesis H1 : SNo y

L1044

Hypothesis H2 : SNo z

L1045

Hypothesis H3 : SNo w

L1046

Hypothesis H4 : w < y

L1047

Hypothesis H5 : SNo (x * y)

L1048

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1049

Hypothesis H7 : SNo (z * y)

L1050

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1051

Hypothesis H9 : SNo (x * w)

L1052

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1053

Hypothesis H11 : SNo (z * w)

L1054

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1055

Hypothesis H13 : SNo (z * y + x * w)

L1056

Hypothesis H14 : SNo (x * y + z * w)

L1057

Hypothesis H15 : SNo u

L1058

Hypothesis H16 : u ∈ SNoL x

L1059

Hypothesis H17 : u ∈ SNoR z

L1060

Hypothesis H18 : SNo (u * y)

L1061

Hypothesis H19 : SNo (u * w)

L1062

Hypothesis H20 : SNo (u * y + x * w)

L1063

Hypothesis H21 : SNo (u * y + z * w)

L1064

Hypothesis H23 : SNo (z * y + u * w)

L1065

Theorem. (Conj_mul_SNo_Lt__54__22)

SNo (z * w + u * y) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 38e09f... and proposition id is be0dfb...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__55__12

L1071

Variable x : set

(*** Conj_mul_SNo_Lt__55__12 TMVLCUjfHPDRPpXfFmXJhzCxQ8KWvLxD4o4 bounty of about 25 bars ***)

L1072

Variable y : set

L1073

Variable z : set

L1074

Variable w : set

L1075

Variable u : set

L1076

Hypothesis H0 : SNo x

L1077

Hypothesis H1 : SNo y

L1078

Hypothesis H2 : SNo z

L1079

Hypothesis H3 : SNo w

L1080

Hypothesis H4 : w < y

L1081

Hypothesis H5 : SNo (x * y)

L1082

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1083

Hypothesis H7 : SNo (z * y)

L1084

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1085

Hypothesis H9 : SNo (x * w)

L1086

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1087

Hypothesis H11 : SNo (z * w)

L1088

Hypothesis H13 : SNo (z * y + x * w)

L1089

Hypothesis H14 : SNo (x * y + z * w)

L1090

Hypothesis H15 : SNo u

L1091

Hypothesis H16 : u ∈ SNoL x

L1092

Hypothesis H17 : u ∈ SNoR z

L1093

Hypothesis H18 : SNo (u * y)

L1094

Hypothesis H19 : SNo (u * w)

L1095

Hypothesis H20 : SNo (u * y + x * w)

L1096

Hypothesis H21 : SNo (u * y + z * w)

L1097

Hypothesis H22 : SNo (x * y + u * w)

L1098

Theorem. (Conj_mul_SNo_Lt__55__12)

SNo (z * y + u * w) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is cad011... and proposition id is 5e4d63...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__55__22

L1104

Variable x : set

(*** Conj_mul_SNo_Lt__55__22 TMUq9F9LLC4ufGCxfaE4goViRAVx1BpDKn1 bounty of about 25 bars ***)

L1105

Variable y : set

L1106

Variable z : set

L1107

Variable w : set

L1108

Variable u : set

L1109

Hypothesis H0 : SNo x

L1110

Hypothesis H1 : SNo y

L1111

Hypothesis H2 : SNo z

L1112

Hypothesis H3 : SNo w

L1113

Hypothesis H4 : w < y

L1114

Hypothesis H5 : SNo (x * y)

L1115

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1116

Hypothesis H7 : SNo (z * y)

L1117

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1118

Hypothesis H9 : SNo (x * w)

L1119

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1120

Hypothesis H11 : SNo (z * w)

L1121

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1122

Hypothesis H13 : SNo (z * y + x * w)

L1123

Hypothesis H14 : SNo (x * y + z * w)

L1124

Hypothesis H15 : SNo u

L1125

Hypothesis H16 : u ∈ SNoL x

L1126

Hypothesis H17 : u ∈ SNoR z

L1127

Hypothesis H18 : SNo (u * y)

L1128

Hypothesis H19 : SNo (u * w)

L1129

Hypothesis H20 : SNo (u * y + x * w)

L1130

Hypothesis H21 : SNo (u * y + z * w)

L1131

Theorem. (Conj_mul_SNo_Lt__55__22)

SNo (z * y + u * w) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is cf5038... and proposition id is aecd5b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__57__1

L1137

Variable x : set

(*** Conj_mul_SNo_Lt__57__1 TMUK2hs55DdmS5maVfUxrqBQAM19SLs5jZQ bounty of about 25 bars ***)

L1138

Variable y : set

L1139

Variable z : set

L1140

Variable w : set

L1141

Variable u : set

L1142

Hypothesis H0 : SNo x

L1143

Hypothesis H2 : SNo z

L1144

Hypothesis H3 : SNo w

L1145

Hypothesis H4 : w < y

L1146

Hypothesis H5 : SNo (x * y)

L1147

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1148

Hypothesis H7 : SNo (z * y)

L1149

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1150

Hypothesis H9 : SNo (x * w)

L1151

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1152

Hypothesis H11 : SNo (z * w)

L1153

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1154

Hypothesis H13 : SNo (z * y + x * w)

L1155

Hypothesis H14 : SNo (x * y + z * w)

L1156

Hypothesis H15 : SNo u

L1157

Hypothesis H16 : u ∈ SNoL x

L1158

Hypothesis H17 : u ∈ SNoR z

L1159

Hypothesis H18 : SNo (u * y)

L1160

Hypothesis H19 : SNo (u * w)

L1161

Hypothesis H20 : SNo (u * y + x * w)

L1162

Theorem. (Conj_mul_SNo_Lt__57__1)

SNo (u * y + z * w) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 76d8ae... and proposition id is ce5ed5...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__57__6

L1168

Variable x : set

(*** Conj_mul_SNo_Lt__57__6 TMHzHyhJY1ERtyzVEHaaWKKsJpYDCuomeVe bounty of about 25 bars ***)

L1169

Variable y : set

L1170

Variable z : set

L1171

Variable w : set

L1172

Variable u : set

L1173

Hypothesis H0 : SNo x

L1174

Hypothesis H1 : SNo y

L1175

Hypothesis H2 : SNo z

L1176

Hypothesis H3 : SNo w

L1177

Hypothesis H4 : w < y

L1178

Hypothesis H5 : SNo (x * y)

L1179

Hypothesis H7 : SNo (z * y)

L1180

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1181

Hypothesis H9 : SNo (x * w)

L1182

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1183

Hypothesis H11 : SNo (z * w)

L1184

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1185

Hypothesis H13 : SNo (z * y + x * w)

L1186

Hypothesis H14 : SNo (x * y + z * w)

L1187

Hypothesis H15 : SNo u

L1188

Hypothesis H16 : u ∈ SNoL x

L1189

Hypothesis H17 : u ∈ SNoR z

L1190

Hypothesis H18 : SNo (u * y)

L1191

Hypothesis H19 : SNo (u * w)

L1192

Hypothesis H20 : SNo (u * y + x * w)

L1193

Theorem. (Conj_mul_SNo_Lt__57__6)

SNo (u * y + z * w) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is be7c12... and proposition id is 24c984...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__57__11

L1199

Variable x : set

(*** Conj_mul_SNo_Lt__57__11 TMThs7inNhfhqiohqCLqCkPeEHEfYnHdNgv bounty of about 25 bars ***)

L1200

Variable y : set

L1201

Variable z : set

L1202

Variable w : set

L1203

Variable u : set

L1204

Hypothesis H0 : SNo x

L1205

Hypothesis H1 : SNo y

L1206

Hypothesis H2 : SNo z

L1207

Hypothesis H3 : SNo w

L1208

Hypothesis H4 : w < y

L1209

Hypothesis H5 : SNo (x * y)

L1210

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1211

Hypothesis H7 : SNo (z * y)

L1212

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1213

Hypothesis H9 : SNo (x * w)

L1214

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1215

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1216

Hypothesis H13 : SNo (z * y + x * w)

L1217

Hypothesis H14 : SNo (x * y + z * w)

L1218

Hypothesis H15 : SNo u

L1219

Hypothesis H16 : u ∈ SNoL x

L1220

Hypothesis H17 : u ∈ SNoR z

L1221

Hypothesis H18 : SNo (u * y)

L1222

Hypothesis H19 : SNo (u * w)

L1223

Hypothesis H20 : SNo (u * y + x * w)

L1224

Theorem. (Conj_mul_SNo_Lt__57__11)

SNo (u * y + z * w) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is e08e78... and proposition id is 989a98...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__60__17

L1230

Variable x : set

(*** Conj_mul_SNo_Lt__60__17 TMPBsLcpDzeBMiXX9YTdTEPQ76fkF2YTpkT bounty of about 25 bars ***)

L1231

Variable y : set

L1232

Variable z : set

L1233

Variable w : set

L1234

Variable u : set

L1235

Hypothesis H0 : SNo x

L1236

Hypothesis H1 : SNo y

L1237

Hypothesis H2 : SNo z

L1238

Hypothesis H3 : SNo w

L1239

Hypothesis H4 : w < y

L1240

Hypothesis H5 : SNo (x * y)

L1241

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1242

Hypothesis H7 : SNo (z * y)

L1243

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1244

Hypothesis H9 : SNo (x * w)

L1245

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1246

Hypothesis H11 : SNo (z * w)

L1247

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1248

Hypothesis H13 : SNo (z * y + x * w)

L1249

Hypothesis H14 : SNo (x * y + z * w)

L1250

Hypothesis H15 : SNo u

L1251

Hypothesis H16 : u ∈ SNoL x

L1252

Theorem. (Conj_mul_SNo_Lt__60__17)

SNo (u * y) → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is 36e597... and proposition id is 2200ac...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_Lt__61__14

L1258

Variable x : set

(*** Conj_mul_SNo_Lt__61__14 TMQLYAVwfyjCuH6gKovkR8B34Sm7pz5hYXf bounty of about 25 bars ***)

L1259

Variable y : set

L1260

Variable z : set

L1261

Variable w : set

L1262

Variable u : set

L1263

Hypothesis H0 : SNo x

L1264

Hypothesis H1 : SNo y

L1265

Hypothesis H2 : SNo z

L1266

Hypothesis H3 : SNo w

L1267

Hypothesis H4 : w < y

L1268

Hypothesis H5 : SNo (x * y)

L1269

Hypothesis H6 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → (v * y + x * x2) < x * y + v * x2))

L1270

Hypothesis H7 : SNo (z * y)

L1271

Hypothesis H8 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoL y → (z * y + v * x2) < v * y + z * x2))

L1272

Hypothesis H9 : SNo (x * w)

L1273

Hypothesis H10 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR w → (x * w + v * x2) < v * w + x * x2))

L1274

Hypothesis H11 : SNo (z * w)

L1275

Hypothesis H12 : (∀v : set, v ∈ SNoR z → (∀x2 : set, x2 ∈ SNoR w → (v * w + z * x2) < z * w + v * x2))

L1276

Hypothesis H13 : SNo (z * y + x * w)

L1277

Hypothesis H15 : SNo u

L1278

Hypothesis H16 : z < u

L1279

Hypothesis H17 : SNoLev u ∈ SNoLev z

L1280

Hypothesis H18 : u ∈ SNoL x

L1281

Theorem. (Conj_mul_SNo_Lt__61__14)

u ∈ SNoR z → (z * y + x * w) < x * y + z * w

In Proofgold the corresponding term root is cedaca... and proposition id is 3c1bf3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__2__2

L1287

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__2__2 TMVAo5ZWMvLqFc9jZqdA3Sn5sTkpaRo1qQP bounty of about 25 bars ***)

L1288

Variable y : set

L1289

Variable z : set

L1290

Variable w : set

L1291

Variable u : set

L1292

Variable v : set

L1293

Hypothesis H0 : SNo x

L1294

Hypothesis H1 : SNo y

L1295

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → (x2 * y + x * y2) < z + x2 * y2))

L1296

Hypothesis H4 : SNo w

L1297

Hypothesis H5 : z < w

L1298

Hypothesis H6 : u ∈ SNoR x

L1299

Hypothesis H7 : v ∈ SNoR y

L1300

Hypothesis H8 : (w + u * v) ≤ u * y + x * v

L1301

Hypothesis H9 : SNo u

L1302

Hypothesis H10 : SNo v

L1303

Hypothesis H11 : SNo (u * v)

L1304

Theorem. (Conj_mul_SNo_SNoL_interpolate__2__2)

(w + u * v) < z + u * v → x * y < w

In Proofgold the corresponding term root is cfae93... and proposition id is 509b2e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__3__1

L1310

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__3__1 TMXKNrjQJLhZVPesgSKot4VkdvEf1VcGtis bounty of about 25 bars ***)

L1311

Variable y : set

L1312

Variable z : set

L1313

Variable w : set

L1314

Variable u : set

L1315

Variable v : set

L1316

Hypothesis H0 : SNo x

L1317

Hypothesis H2 : SNo z

L1318

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → (x2 * y + x * y2) < z + x2 * y2))

L1319

Hypothesis H4 : SNo w

L1320

Hypothesis H5 : z < w

L1321

Hypothesis H6 : u ∈ SNoR x

L1322

Hypothesis H7 : v ∈ SNoR y

L1323

Hypothesis H8 : (w + u * v) ≤ u * y + x * v

L1324

Hypothesis H9 : SNo u

L1325

Hypothesis H10 : SNo v

L1326

Theorem. (Conj_mul_SNo_SNoL_interpolate__3__1)

SNo (u * v) → x * y < w

In Proofgold the corresponding term root is cf24d5... and proposition id is 70acb8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__3__2

L1332

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__3__2 TMHTnDXkxhmMZHqcTRkaPapaxQ6tAWz1kFC bounty of about 25 bars ***)

L1333

Variable y : set

L1334

Variable z : set

L1335

Variable w : set

L1336

Variable u : set

L1337

Variable v : set

L1338

Hypothesis H0 : SNo x

L1339

Hypothesis H1 : SNo y

L1340

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → (x2 * y + x * y2) < z + x2 * y2))

L1341

Hypothesis H4 : SNo w

L1342

Hypothesis H5 : z < w

L1343

Hypothesis H6 : u ∈ SNoR x

L1344

Hypothesis H7 : v ∈ SNoR y

L1345

Hypothesis H8 : (w + u * v) ≤ u * y + x * v

L1346

Hypothesis H9 : SNo u

L1347

Hypothesis H10 : SNo v

L1348

Theorem. (Conj_mul_SNo_SNoL_interpolate__3__2)

SNo (u * v) → x * y < w

In Proofgold the corresponding term root is 0401c9... and proposition id is b91486...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__3__5

L1354

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__3__5 TMTHJ61q2mNUAPwUh9k478498hw3C9rZfAH bounty of about 25 bars ***)

L1355

Variable y : set

L1356

Variable z : set

L1357

Variable w : set

L1358

Variable u : set

L1359

Variable v : set

L1360

Hypothesis H0 : SNo x

L1361

Hypothesis H1 : SNo y

L1362

Hypothesis H2 : SNo z

L1363

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → (x2 * y + x * y2) < z + x2 * y2))

L1364

Hypothesis H4 : SNo w

L1365

Hypothesis H6 : u ∈ SNoR x

L1366

Hypothesis H7 : v ∈ SNoR y

L1367

Hypothesis H8 : (w + u * v) ≤ u * y + x * v

L1368

Hypothesis H9 : SNo u

L1369

Hypothesis H10 : SNo v

L1370

Theorem. (Conj_mul_SNo_SNoL_interpolate__3__5)

SNo (u * v) → x * y < w

In Proofgold the corresponding term root is 743c37... and proposition id is b71763...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__5__9

L1376

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__5__9 TMcAaTaV3H6diAjXiBbRqoCxFZTknJNeJ3F bounty of about 25 bars ***)

L1377

Variable y : set

L1378

Variable z : set

L1379

Variable w : set

L1380

Variable u : set

L1381

Variable v : set

L1382

Hypothesis H0 : SNo x

L1383

Hypothesis H1 : SNo y

L1384

Hypothesis H2 : SNo z

L1385

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → (x2 * y + x * y2) < z + x2 * y2))

L1386

Hypothesis H4 : SNo w

L1387

Hypothesis H5 : z < w

L1388

Hypothesis H6 : u ∈ SNoL x

L1389

Hypothesis H7 : v ∈ SNoL y

L1390

Hypothesis H8 : (w + u * v) ≤ u * y + x * v

L1391

Hypothesis H10 : SNo v

L1392

Hypothesis H11 : SNo (u * v)

L1393

Theorem. (Conj_mul_SNo_SNoL_interpolate__5__9)

(w + u * v) < z + u * v → x * y < w

In Proofgold the corresponding term root is 25b05d... and proposition id is 5101ed...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__7__3

L1399

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__7__3 TMYThaUfhYWi7j8ULQjaP3WuFToo399mUmJ bounty of about 25 bars ***)

L1400

Variable y : set

L1401

Variable z : set

L1402

Variable w : set

L1403

Hypothesis H0 : SNo x

L1404

Hypothesis H1 : SNo y

L1405

Hypothesis H2 : SNo (x * y)

L1406

Hypothesis H4 : (∀u : set, u ∈ SNoS_ (SNoLev z) → SNoLev u ∈ SNoLev (x * y) → u < x * y → (∃v : set, v ∈ SNoL x ∧ (∃x2 : set, x2 ∈ SNoL y ∧ (u + v * x2) ≤ v * y + x * x2)) ∨ (∃v : set, v ∈ SNoR x ∧ (∃x2 : set, x2 ∈ SNoR y ∧ (u + v * x2) ≤ v * y + x * x2)))

L1407

Hypothesis H5 : SNoLev z ∈ SNoLev (x * y)

L1408

Hypothesis H6 : (∀u : set, u ∈ SNoL x → (∀v : set, v ∈ SNoL y → (u * y + x * v) < z + u * v))

L1409

Hypothesis H7 : (∀u : set, u ∈ SNoR x → (∀v : set, v ∈ SNoR y → (u * y + x * v) < z + u * v))

L1410

Hypothesis H8 : SNo w

L1411

Hypothesis H9 : SNoLev w ∈ SNoLev z

L1412

Hypothesis H10 : z < w

L1413

Hypothesis H11 : w < x * y

L1414

Theorem. (Conj_mul_SNo_SNoL_interpolate__7__3)

(∃u : set, u ∈ SNoL x ∧ (∃v : set, v ∈ SNoL y ∧ (w + u * v) ≤ u * y + x * v)) ∨ (∃u : set, u ∈ SNoR x ∧ (∃v : set, v ∈ SNoR y ∧ (w + u * v) ≤ u * y + x * v)) → x * y < w

In Proofgold the corresponding term root is 206647... and proposition id is 98f4ce...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__8__1

L1420

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__8__1 TMMLhZJbXGaEPxjZrJHX6DXHPi4oJMypbSW bounty of about 25 bars ***)

L1421

Variable y : set

L1422

Variable z : set

L1423

Hypothesis H0 : SNo x

L1424

Hypothesis H2 : SNo (x * y)

L1425

Hypothesis H3 : SNo z

L1426

Hypothesis H4 : (∀w : set, w ∈ SNoS_ (SNoLev z) → SNoLev w ∈ SNoLev (x * y) → w < x * y → (∃u : set, u ∈ SNoL x ∧ (∃v : set, v ∈ SNoL y ∧ (w + u * v) ≤ u * y + x * v)) ∨ (∃u : set, u ∈ SNoR x ∧ (∃v : set, v ∈ SNoR y ∧ (w + u * v) ≤ u * y + x * v)))

L1427

Hypothesis H5 : SNoLev z ∈ SNoLev (x * y)

L1428

Hypothesis H6 : z < x * y

L1429

Hypothesis H7 : ¬ ((∃w : set, w ∈ SNoL x ∧ (∃u : set, u ∈ SNoL y ∧ (z + w * u) ≤ w * y + x * u)) ∨ (∃w : set, w ∈ SNoR x ∧ (∃u : set, u ∈ SNoR y ∧ (z + w * u) ≤ w * y + x * u)))

L1430

Hypothesis H8 : (∀w : set, w ∈ SNoL x → (∀u : set, u ∈ SNoL y → (w * y + x * u) < z + w * u))

L1431

Theorem. (Conj_mul_SNo_SNoL_interpolate__8__1)

¬ (∀w : set, w ∈ SNoR x → (∀u : set, u ∈ SNoR y → (w * y + x * u) < z + w * u))

In Proofgold the corresponding term root is af7aac... and proposition id is b6f48a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__9__4

L1437

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__9__4 TMYRPdKdbZhUiknHS6qmv9PKXmDTFut2jz3 bounty of about 25 bars ***)

L1438

Variable y : set

L1439

Variable z : set

L1440

Hypothesis H0 : SNo x

L1441

Hypothesis H1 : SNo y

L1442

Hypothesis H2 : SNo (x * y)

L1443

Hypothesis H3 : SNo z

L1444

Hypothesis H5 : SNoLev z ∈ SNoLev (x * y)

L1445

Hypothesis H6 : z < x * y

L1446

L1447

Theorem. (Conj_mul_SNo_SNoL_interpolate__9__4)

¬ (∀w : set, w ∈ SNoL x → (∀u : set, u ∈ SNoL y → (w * y + x * u) < z + w * u))

In Proofgold the corresponding term root is 184d4f... and proposition id is 5d8b86...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoL_interpolate__11__0

L1453

Variable x : set

(*** Conj_mul_SNo_SNoL_interpolate__11__0 TMYpZAftBM1Y86VtuXn7zSxUkXbrXf5LFrL bounty of about 25 bars ***)

L1454

Variable y : set

L1455

Hypothesis H1 : SNo y

L1456

Theorem. (Conj_mul_SNo_SNoL_interpolate__11__0)

SNo (x * y) → (∀z : set, z ∈ SNoL (x * y) → (∃w : set, w ∈ SNoL x ∧ (∃u : set, u ∈ SNoL y ∧ (z + w * u) ≤ w * y + x * u)) ∨ (∃w : set, w ∈ SNoR x ∧ (∃u : set, u ∈ SNoR y ∧ (z + w * u) ≤ w * y + x * u)))

In Proofgold the corresponding term root is 61218f... and proposition id is 326888...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__1__3

L1462

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__1__3 TMTW3d62qPEXJRDbjEhWrPo2VjqLBnT9wyH bounty of about 25 bars ***)

L1463

Variable y : set

L1464

Variable z : set

L1465

Variable w : set

L1466

Variable u : set

L1467

Variable v : set

L1468

Hypothesis H0 : SNo z

L1469

Hypothesis H1 : SNo w

L1470

Hypothesis H2 : w < z

L1471

Hypothesis H4 : (z + u * v) < w + u * v

L1472

Theorem. (Conj_mul_SNo_SNoR_interpolate__1__3)

z < w → w < x * y

In Proofgold the corresponding term root is ac1d32... and proposition id is 3cff72...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__3__1

L1478

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__3__1 TMGJ7YT7JCpjHMUNfiMFirSniXjTzj58jCE bounty of about 25 bars ***)

L1479

Variable y : set

L1480

Variable z : set

L1481

Variable w : set

L1482

Variable u : set

L1483

Variable v : set

L1484

Hypothesis H0 : SNo x

L1485

Hypothesis H2 : SNo z

L1486

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → (z + x2 * y2) < x2 * y + x * y2))

L1487

Hypothesis H4 : SNo w

L1488

Hypothesis H5 : w < z

L1489

Hypothesis H6 : u ∈ SNoR x

L1490

Hypothesis H7 : v ∈ SNoL y

L1491

Hypothesis H8 : (u * y + x * v) ≤ w + u * v

L1492

Hypothesis H9 : SNo u

L1493

Hypothesis H10 : SNo v

L1494

Theorem. (Conj_mul_SNo_SNoR_interpolate__3__1)

SNo (u * v) → w < x * y

In Proofgold the corresponding term root is d41b52... and proposition id is 8be274...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__3__8

L1500

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__3__8 TMKCe66ZKMpabiCDcttkP56VezaV4ytgTai bounty of about 25 bars ***)

L1501

Variable y : set

L1502

Variable z : set

L1503

Variable w : set

L1504

Variable u : set

L1505

Variable v : set

L1506

Hypothesis H0 : SNo x

L1507

Hypothesis H1 : SNo y

L1508

Hypothesis H2 : SNo z

L1509

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → (z + x2 * y2) < x2 * y + x * y2))

L1510

Hypothesis H4 : SNo w

L1511

Hypothesis H5 : w < z

L1512

Hypothesis H6 : u ∈ SNoR x

L1513

Hypothesis H7 : v ∈ SNoL y

L1514

Hypothesis H9 : SNo u

L1515

Hypothesis H10 : SNo v

L1516

Theorem. (Conj_mul_SNo_SNoR_interpolate__3__8)

SNo (u * v) → w < x * y

In Proofgold the corresponding term root is 9ad7f7... and proposition id is fbb77a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__4__3

L1522

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__4__3 TMTW3d62qPEXJRDbjEhWrPo2VjqLBnT9wyH bounty of about 25 bars ***)

L1523

Variable y : set

L1524

Variable z : set

L1525

Variable w : set

L1526

Variable u : set

L1527

Variable v : set

L1528

Hypothesis H0 : SNo z

L1529

Hypothesis H1 : SNo w

L1530

Hypothesis H2 : w < z

L1531

Hypothesis H4 : (z + u * v) < w + u * v

L1532

Theorem. (Conj_mul_SNo_SNoR_interpolate__4__3)

z < w → w < x * y

In Proofgold the corresponding term root is ac1d32... and proposition id is 3cff72...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__6__4

L1538

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__6__4 TMSdXhaLYDDfeQTgWG5cbwWGmaGMp1ZvAL4 bounty of about 25 bars ***)

L1539

Variable y : set

L1540

Variable z : set

L1541

Variable w : set

L1542

Variable u : set

L1543

Variable v : set

L1544

Hypothesis H0 : SNo x

L1545

Hypothesis H1 : SNo y

L1546

Hypothesis H2 : SNo z

L1547

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR y → (z + x2 * y2) < x2 * y + x * y2))

L1548

Hypothesis H5 : w < z

L1549

Hypothesis H6 : u ∈ SNoL x

L1550

Hypothesis H7 : v ∈ SNoR y

L1551

Hypothesis H8 : (u * y + x * v) ≤ w + u * v

L1552

Hypothesis H9 : SNo u

L1553

Hypothesis H10 : SNo v

L1554

Theorem. (Conj_mul_SNo_SNoR_interpolate__6__4)

SNo (u * v) → w < x * y

In Proofgold the corresponding term root is 7d9860... and proposition id is 912f6c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__7__3

L1560

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__7__3 TMXVhXyGsXGLDb8MGn2yuenoebtuYChchYH bounty of about 25 bars ***)

L1561

Variable y : set

L1562

Variable z : set

L1563

Variable w : set

L1564

Hypothesis H0 : SNo x

L1565

Hypothesis H1 : SNo y

L1566

Hypothesis H2 : SNo (x * y)

L1567

Hypothesis H4 : (∀u : set, u ∈ SNoS_ (SNoLev z) → SNoLev u ∈ SNoLev (x * y) → x * y < u → (∃v : set, v ∈ SNoL x ∧ (∃x2 : set, x2 ∈ SNoR y ∧ (v * y + x * x2) ≤ u + v * x2)) ∨ (∃v : set, v ∈ SNoR x ∧ (∃x2 : set, x2 ∈ SNoL y ∧ (v * y + x * x2) ≤ u + v * x2)))

L1568

Hypothesis H5 : SNoLev z ∈ SNoLev (x * y)

L1569

Hypothesis H6 : (∀u : set, u ∈ SNoL x → (∀v : set, v ∈ SNoR y → (z + u * v) < u * y + x * v))

L1570

Hypothesis H7 : (∀u : set, u ∈ SNoR x → (∀v : set, v ∈ SNoL y → (z + u * v) < u * y + x * v))

L1571

Hypothesis H8 : SNo w

L1572

Hypothesis H9 : SNoLev w ∈ SNoLev z

L1573

Hypothesis H10 : w < z

L1574

Hypothesis H11 : x * y < w

L1575

Theorem. (Conj_mul_SNo_SNoR_interpolate__7__3)

(∃u : set, u ∈ SNoL x ∧ (∃v : set, v ∈ SNoR y ∧ (u * y + x * v) ≤ w + u * v)) ∨ (∃u : set, u ∈ SNoR x ∧ (∃v : set, v ∈ SNoL y ∧ (u * y + x * v) ≤ w + u * v)) → w < x * y

In Proofgold the corresponding term root is d39608... and proposition id is 642030...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_SNoR_interpolate__8__2

L1581

Variable x : set

(*** Conj_mul_SNo_SNoR_interpolate__8__2 TMZeuFaoPT4P9adgbZz1UNoxhEq4Pa3WfzE bounty of about 25 bars ***)

L1582

Variable y : set

L1583

Variable z : set

L1584

Hypothesis H0 : SNo x

L1585

Hypothesis H1 : SNo y

L1586

Hypothesis H3 : SNo z

L1587

Hypothesis H4 : (∀w : set, w ∈ SNoS_ (SNoLev z) → SNoLev w ∈ SNoLev (x * y) → x * y < w → (∃u : set, u ∈ SNoL x ∧ (∃v : set, v ∈ SNoR y ∧ (u * y + x * v) ≤ w + u * v)) ∨ (∃u : set, u ∈ SNoR x ∧ (∃v : set, v ∈ SNoL y ∧ (u * y + x * v) ≤ w + u * v)))

L1588

Hypothesis H5 : SNoLev z ∈ SNoLev (x * y)

L1589

Hypothesis H6 : x * y < z

L1590

Hypothesis H7 : ¬ ((∃w : set, w ∈ SNoL x ∧ (∃u : set, u ∈ SNoR y ∧ (w * y + x * u) ≤ z + w * u)) ∨ (∃w : set, w ∈ SNoR x ∧ (∃u : set, u ∈ SNoL y ∧ (w * y + x * u) ≤ z + w * u)))

L1591

Hypothesis H8 : (∀w : set, w ∈ SNoL x → (∀u : set, u ∈ SNoR y → (z + w * u) < w * y + x * u))

L1592

Theorem. (Conj_mul_SNo_SNoR_interpolate__8__2)

¬ (∀w : set, w ∈ SNoR x → (∀u : set, u ∈ SNoL y → (z + w * u) < w * y + x * u))

In Proofgold the corresponding term root is babb0e... and proposition id is 69be7a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_oneR__3__0

L1598

Variable x : set

(*** Conj_mul_SNo_oneR__3__0 TMFB5yf5a9bzsSVSHhCx8mPBPvPQkR4AvXY bounty of about 25 bars ***)

L1599

Variable y : set

L1600

Hypothesis H1 : (∀z : set, z ∈ SNoL x → (∀w : set, w ∈ SNoL (ordsucc Empty) → (z * ordsucc Empty + x * w) < x * ordsucc Empty + z * w))

L1601

Hypothesis H2 : Empty ∈ SNoL (ordsucc Empty)

L1602

Hypothesis H3 : y ∈ SNoL x

L1603

Hypothesis H4 : SNo y

L1604

Hypothesis H5 : y * ordsucc Empty + x * Empty = y

L1605

Theorem. (Conj_mul_SNo_oneR__3__0)

x * ordsucc Empty + y * Empty = x * ordsucc Empty → y < x * ordsucc Empty

In Proofgold the corresponding term root is 078312... and proposition id is 59fb31...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_com__1__0

L1611

Variable x : set

(*** Conj_mul_SNo_com__1__0 TMMWSnpFz5mJNhwYF4osH1UqKFrztVXEe9V bounty of about 25 bars ***)

L1612

Variable y : set

L1613

Variable z : set

L1614

Variable w : set

L1615

Variable u : set

L1616

Variable v : set

L1617

Hypothesis H1 : SNo y

L1618

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → x2 * y = y * x2)

L1619

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → x * x2 = x2 * x)

L1620

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x2 * y2 = y2 * x2))

L1621

Hypothesis H5 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))

L1622

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1623

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1624

Hypothesis H8 : (∀x2 : set, x2 ∈ v → (∀P : prop, (∀y2 : set, y2 ∈ SNoL y → (∀z2 : set, z2 ∈ SNoR x → x2 = y2 * x + y * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR y → (∀z2 : set, z2 ∈ SNoL x → x2 = y2 * x + y * z2 + - (y2 * z2) → P)) → P))

L1625

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1626

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1627

Hypothesis H11 : z = u

L1628

Theorem. (Conj_mul_SNo_com__1__0)

w = v → SNoCut z w = SNoCut u v

In Proofgold the corresponding term root is 107a29... and proposition id is 7b761e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_com__2__5

L1634

Variable x : set

(*** Conj_mul_SNo_com__2__5 TMQBn2Vmh5519uKLgvALG1RLir74uKm6kNN bounty of about 25 bars ***)

L1635

Variable y : set

L1636

Variable z : set

L1637

Variable w : set

L1638

Variable u : set

L1639

Variable v : set

L1640

Hypothesis H0 : SNo x

L1641

Hypothesis H1 : SNo y

L1642

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → x2 * y = y * x2)

L1643

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → x * x2 = x2 * x)

L1644

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x2 * y2 = y2 * x2))

L1645

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1646

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1647

Hypothesis H8 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))

L1648

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1649

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1650

Hypothesis H11 : (∀x2 : set, x2 ∈ u → (∀P : prop, (∀y2 : set, y2 ∈ SNoL y → (∀z2 : set, z2 ∈ SNoL x → x2 = y2 * x + y * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR y → (∀z2 : set, z2 ∈ SNoR x → x2 = y2 * x + y * z2 + - (y2 * z2) → P)) → P))

L1651

Hypothesis H12 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1652

Hypothesis H13 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1653

Hypothesis H14 : (∀x2 : set, x2 ∈ v → (∀P : prop, (∀y2 : set, y2 ∈ SNoL y → (∀z2 : set, z2 ∈ SNoR x → x2 = y2 * x + y * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR y → (∀z2 : set, z2 ∈ SNoL x → x2 = y2 * x + y * z2 + - (y2 * z2) → P)) → P))

L1654

Hypothesis H15 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1655

Hypothesis H16 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1656

Theorem. (Conj_mul_SNo_com__2__5)

z = u → SNoCut z w = SNoCut u v

In Proofgold the corresponding term root is 886af3... and proposition id is 9f7679...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_com__2__9

L1662

Variable x : set

(*** Conj_mul_SNo_com__2__9 TMXzBz7GFYvBMLPG62PoH8ymNGTBA1uUE5b bounty of about 25 bars ***)

L1663

Variable y : set

L1664

Variable z : set

L1665

Variable w : set

L1666

Variable u : set

L1667

Variable v : set

L1668

Hypothesis H0 : SNo x

L1669

Hypothesis H1 : SNo y

L1670

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → x2 * y = y * x2)

L1671

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → x * x2 = x2 * x)

L1672

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x2 * y2 = y2 * x2))

L1673

Hypothesis H5 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))

L1674

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1675

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1676

L1677

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1678

L1679

Hypothesis H12 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1680

Hypothesis H13 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1681

L1682

Hypothesis H15 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1683

Hypothesis H16 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1684

Theorem. (Conj_mul_SNo_com__2__9)

z = u → SNoCut z w = SNoCut u v

In Proofgold the corresponding term root is 9d3a0e... and proposition id is 462c54...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_com__2__11

L1690

Variable x : set

(*** Conj_mul_SNo_com__2__11 TMUdwb2A7HxFGAMvBzeMdz85SUezBqpZWx3 bounty of about 25 bars ***)

L1691

Variable y : set

L1692

Variable z : set

L1693

Variable w : set

L1694

Variable u : set

L1695

Variable v : set

L1696

Hypothesis H0 : SNo x

L1697

Hypothesis H1 : SNo y

L1698

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → x2 * y = y * x2)

L1699

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → x * x2 = x2 * x)

L1700

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x2 * y2 = y2 * x2))

L1701

L1702

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1703

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1704

L1705

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1706

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1707

Hypothesis H12 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1708

Hypothesis H13 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1709

L1710

Hypothesis H15 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1711

Hypothesis H16 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1712

Theorem. (Conj_mul_SNo_com__2__11)

z = u → SNoCut z w = SNoCut u v

In Proofgold the corresponding term root is c48471... and proposition id is bfee04...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_com__2__14

L1718

Variable x : set

(*** Conj_mul_SNo_com__2__14 TMW92VQFJRVgNVRwkSTXVPYCVu56UJc1hDA bounty of about 25 bars ***)

L1719

Variable y : set

L1720

Variable z : set

L1721

Variable w : set

L1722

Variable u : set

L1723

Variable v : set

L1724

Hypothesis H0 : SNo x

L1725

Hypothesis H1 : SNo y

L1726

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → x2 * y = y * x2)

L1727

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → x * x2 = x2 * x)

L1728

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x2 * y2 = y2 * x2))

L1729

L1730

Hypothesis H6 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1731

Hypothesis H7 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ z))

L1732

L1733

Hypothesis H9 : (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1734

Hypothesis H10 : (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL y → x2 * y + x * y2 + - (x2 * y2) ∈ w))

L1735

L1736

Hypothesis H12 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1737

Hypothesis H13 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ u))

L1738

Hypothesis H15 : (∀x2 : set, x2 ∈ SNoL y → (∀y2 : set, y2 ∈ SNoR x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1739

Hypothesis H16 : (∀x2 : set, x2 ∈ SNoR y → (∀y2 : set, y2 ∈ SNoL x → x2 * x + y * y2 + - (x2 * y2) ∈ v))

L1740

Theorem. (Conj_mul_SNo_com__2__14)

z = u → SNoCut z w = SNoCut u v

In Proofgold the corresponding term root is 56cd8b... and proposition id is 87b290...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__2__12

L1746

Variable x : set

(*** Conj_mul_SNo_minus_distrL__2__12 TMEuc4utY3sCFs4zY5edG8aHEUqzqmWAkNW bounty of about 25 bars ***)

L1747

Variable y : set

L1748

Variable z : set

L1749

Variable w : set

L1750

Variable u : set

L1751

Variable v : set

L1752

Hypothesis H0 : SNo x

L1753

Hypothesis H1 : SNo y

L1754

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → - x2 * y = - (x2 * y))

L1755

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - x * x2 = - (x * x2))

L1756

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → - x2 * y2 = - (x2 * y2)))

L1757

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL (- x) → (∀y2 : set, y2 ∈ SNoR y → x2 * y + - x * y2 + - (x2 * y2) ∈ z))

L1758

Hypothesis H6 : u ∈ SNoR x

L1759

Hypothesis H7 : v ∈ SNoR y

L1760

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

L1761

Hypothesis H9 : SNo u

L1762

Hypothesis H10 : SNoLev u ∈ SNoLev x

L1763

Hypothesis H11 : x < u

L1764

Hypothesis H13 : SNo (- u)

L1765

Theorem. (Conj_mul_SNo_minus_distrL__2__12)

- u ∈ SNoL (- x) → - w ∈ z

In Proofgold the corresponding term root is e53912... and proposition id is fc161e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__4__7

L1771

Variable x : set

(*** Conj_mul_SNo_minus_distrL__4__7 TMMZ866TgcntfrQnfcE1aV3fQRbFoZxXnXL bounty of about 25 bars ***)

L1772

Variable y : set

L1773

Variable z : set

L1774

Variable w : set

L1775

Variable u : set

L1776

Variable v : set

L1777

Hypothesis H0 : SNo x

L1778

Hypothesis H1 : SNo y

L1779

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → - x2 * y = - (x2 * y))

L1780

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - x * x2 = - (x * x2))

L1781

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → - x2 * y2 = - (x2 * y2)))

L1782

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoR (- x) → (∀y2 : set, y2 ∈ SNoL y → x2 * y + - x * y2 + - (x2 * y2) ∈ z))

L1783

Hypothesis H6 : u ∈ SNoL x

L1784

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

L1785

Hypothesis H9 : SNo u

L1786

Hypothesis H10 : SNo v

L1787

Hypothesis H11 : - u ∈ SNoR (- x)

L1788

Theorem. (Conj_mul_SNo_minus_distrL__4__7)

- w = - u * y + - x * v + - (- u * v) → - w ∈ z

In Proofgold the corresponding term root is 3158f3... and proposition id is 9f19d9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__5__1

L1794

Variable x : set

(*** Conj_mul_SNo_minus_distrL__5__1 TMYB24ggbKYc7vT31MvKD1avEN6jGZGsgbp bounty of about 25 bars ***)

L1795

Variable y : set

L1796

Variable z : set

L1797

Variable w : set

L1798

Variable u : set

L1799

Variable v : set

L1800

Hypothesis H0 : SNo x

L1801

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → - x2 * y = - (x2 * y))

L1802

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - x * x2 = - (x * x2))

L1803

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → - x2 * y2 = - (x2 * y2)))

L1804

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoR (- x) → (∀y2 : set, y2 ∈ SNoL y → x2 * y + - x * y2 + - (x2 * y2) ∈ z))

L1805

Hypothesis H6 : u ∈ SNoL x

L1806

Hypothesis H7 : v ∈ SNoL y

L1807

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

L1808

Hypothesis H9 : SNo u

L1809

Hypothesis H10 : SNoLev u ∈ SNoLev x

L1810

Hypothesis H11 : u < x

L1811

Hypothesis H12 : SNo v

L1812

Hypothesis H13 : SNo (- u)

L1813

Theorem. (Conj_mul_SNo_minus_distrL__5__1)

- u ∈ SNoR (- x) → - w ∈ z

In Proofgold the corresponding term root is be881d... and proposition id is af92e4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__9__0

L1819

Variable x : set

(*** Conj_mul_SNo_minus_distrL__9__0 TMcb5AsUczGPnddJ5Rugfq8T6dxRuzhKKs5 bounty of about 25 bars ***)

L1820

Variable y : set

L1821

Variable z : set

L1822

Variable w : set

L1823

Variable u : set

L1824

Hypothesis H1 : SNo y

L1825

Hypothesis H2 : (∀v : set, v ∈ SNoS_ (SNoLev x) → - v * y = - (v * y))

L1826

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev y) → - x * v = - (x * v))

L1827

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - v * x2 = - (v * x2)))

L1828

Hypothesis H5 : (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → v * y + x * x2 + - (v * x2) ∈ z))

L1829

Hypothesis H6 : u ∈ SNoL y

L1830

Hypothesis H7 : SNo w

L1831

Hypothesis H8 : SNoLev w ∈ SNoLev (- x)

L1832

Hypothesis H9 : - x < w

L1833

Hypothesis H10 : SNo u

L1834

Theorem. (Conj_mul_SNo_minus_distrL__9__0)

SNo (- w) → w * y + - x * u + - (w * u) ∈ Repl z minus_SNo

In Proofgold the corresponding term root is a81fc3... and proposition id is 3319ae...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__15__11

L1840

Variable x : set

(*** Conj_mul_SNo_minus_distrL__15__11 TMKFZcuy6dZKmXM9DjVgWxfjLXrBwU8toPP bounty of about 25 bars ***)

L1841

Variable y : set

L1842

Variable z : set

L1843

Variable w : set

L1844

Variable u : set

L1845

Variable v : set

L1846

Hypothesis H0 : SNo x

L1847

Hypothesis H1 : SNo y

L1848

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → - x2 * y = - (x2 * y))

L1849

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - x * x2 = - (x * x2))

L1850

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → - x2 * y2 = - (x2 * y2)))

L1851

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL (- x) → (∀y2 : set, y2 ∈ SNoL y → x2 * y + - x * y2 + - (x2 * y2) ∈ z))

L1852

Hypothesis H6 : u ∈ SNoR x

L1853

Hypothesis H7 : v ∈ SNoL y

L1854

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

L1855

Hypothesis H9 : SNo u

L1856

Hypothesis H10 : SNoLev u ∈ SNoLev x

L1857

Hypothesis H12 : SNo v

L1858

Theorem. (Conj_mul_SNo_minus_distrL__15__11)

SNo (- u) → - w ∈ z

In Proofgold the corresponding term root is 2859a0... and proposition id is a497a9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__18__8

L1864

Variable x : set

(*** Conj_mul_SNo_minus_distrL__18__8 TMQxAaFM64wyeSLR7B2RG8MhUvJvhfuGpFS bounty of about 25 bars ***)

L1865

Variable y : set

L1866

Variable z : set

L1867

Variable w : set

L1868

Variable u : set

L1869

Variable v : set

L1870

Hypothesis H0 : SNo x

L1871

Hypothesis H1 : SNo y

L1872

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → - x2 * y = - (x2 * y))

L1873

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - x * x2 = - (x * x2))

L1874

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → - x2 * y2 = - (x2 * y2)))

L1875

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoR (- x) → (∀y2 : set, y2 ∈ SNoR y → x2 * y + - x * y2 + - (x2 * y2) ∈ z))

L1876

Hypothesis H6 : u ∈ SNoL x

L1877

Hypothesis H7 : v ∈ SNoR y

L1878

Hypothesis H9 : SNo u

L1879

Hypothesis H10 : SNoLev u ∈ SNoLev x

L1880

Hypothesis H11 : u < x

L1881

Hypothesis H12 : SNo v

L1882

Theorem. (Conj_mul_SNo_minus_distrL__18__8)

SNo (- u) → - w ∈ z

In Proofgold the corresponding term root is 270ba7... and proposition id is 72d02e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__18__10

L1888

Variable x : set

(*** Conj_mul_SNo_minus_distrL__18__10 TMKAHt92bfoRAcp3VWJ2TLjVhUgbaxYBJaX bounty of about 25 bars ***)

L1889

Variable y : set

L1890

Variable z : set

L1891

Variable w : set

L1892

Variable u : set

L1893

Variable v : set

L1894

Hypothesis H0 : SNo x

L1895

Hypothesis H1 : SNo y

L1896

Hypothesis H2 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → - x2 * y = - (x2 * y))

L1897

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - x * x2 = - (x * x2))

L1898

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → - x2 * y2 = - (x2 * y2)))

L1899

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoR (- x) → (∀y2 : set, y2 ∈ SNoR y → x2 * y + - x * y2 + - (x2 * y2) ∈ z))

L1900

Hypothesis H6 : u ∈ SNoL x

L1901

Hypothesis H7 : v ∈ SNoR y

L1902

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

L1903

Hypothesis H9 : SNo u

L1904

Hypothesis H11 : u < x

L1905

Hypothesis H12 : SNo v

L1906

Theorem. (Conj_mul_SNo_minus_distrL__18__10)

SNo (- u) → - w ∈ z

In Proofgold the corresponding term root is 2fb0e9... and proposition id is 3511ac...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__19__5

L1912

Variable x : set

(*** Conj_mul_SNo_minus_distrL__19__5 TMbMyDKcru4qMnGZodvvSAUZMssyJhm5Tqf bounty of about 25 bars ***)

L1913

Variable y : set

L1914

Variable z : set

L1915

Variable w : set

L1916

Variable u : set

L1917

Hypothesis H0 : SNo x

L1918

Hypothesis H1 : SNo y

L1919

Hypothesis H2 : (∀v : set, v ∈ SNoS_ (SNoLev x) → - v * y = - (v * y))

L1920

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev y) → - x * v = - (x * v))

L1921

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - v * x2 = - (v * x2)))

L1922

Hypothesis H6 : u ∈ SNoR y

L1923

Hypothesis H7 : SNo w

L1924

Hypothesis H8 : SNo u

L1925

Hypothesis H9 : SNo (- w)

L1926

Hypothesis H10 : - w ∈ SNoL x

L1927

Theorem. (Conj_mul_SNo_minus_distrL__19__5)

w * y + - x * u + - (w * u) = - (- w * y + x * u + - (- w * u)) → w * y + - x * u + - (w * u) ∈ Repl z minus_SNo

In Proofgold the corresponding term root is e43d51... and proposition id is e3deb5...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__22__7

L1933

Variable x : set

(*** Conj_mul_SNo_minus_distrL__22__7 TMH8okuPDoASFJE1vY5pm2ws9K5VsGbvkbJ bounty of about 25 bars ***)

L1934

Variable y : set

L1935

Variable z : set

L1936

Variable w : set

L1937

Variable u : set

L1938

Hypothesis H0 : SNo x

L1939

Hypothesis H1 : SNo y

L1940

Hypothesis H2 : (∀v : set, v ∈ SNoS_ (SNoLev x) → - v * y = - (v * y))

L1941

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev y) → - x * v = - (x * v))

L1942

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → - v * x2 = - (v * x2)))

L1943

Hypothesis H5 : (∀v : set, v ∈ SNoR x → (∀x2 : set, x2 ∈ SNoL y → v * y + x * x2 + - (v * x2) ∈ z))

L1944

Hypothesis H6 : u ∈ SNoL y

L1945

Hypothesis H8 : SNo u

L1946

Hypothesis H9 : SNo (- w)

L1947

Hypothesis H10 : - w ∈ SNoR x

L1948

Theorem. (Conj_mul_SNo_minus_distrL__22__7)

w * y + - x * u + - (w * u) = - (- w * y + x * u + - (- w * u)) → w * y + - x * u + - (w * u) ∈ Repl z minus_SNo

In Proofgold the corresponding term root is 117227... and proposition id is 49b4ac...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_minus_distrL__26__0

L1954

Variable x : set

(*** Conj_mul_SNo_minus_distrL__26__0 TMNdi4TQGs861uU8DhP7Da8HDTnpKXc8CZv bounty of about 25 bars ***)

L1955

Variable y : set

L1956

Hypothesis H1 : SNo y

L1957

Hypothesis H2 : (∀z : set, z ∈ SNoS_ (SNoLev x) → - z * y = - (z * y))

L1958

Hypothesis H3 : (∀z : set, z ∈ SNoS_ (SNoLev y) → - x * z = - (x * z))

L1959

Hypothesis H4 : (∀z : set, z ∈ SNoS_ (SNoLev x) → (∀w : set, w ∈ SNoS_ (SNoLev y) → - z * w = - (z * w)))

L1960

Theorem. (Conj_mul_SNo_minus_distrL__26__0)

SNo (- x) → - x * y = - (x * y)

In Proofgold the corresponding term root is 203f35... and proposition id is 022d7e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__1__23

L1966

Variable x : set

(*** Conj_mul_SNo_distrR__1__23 TMU94X5sAr9MxqwdNdm78WufsjbLhU6Cdjt bounty of about 25 bars ***)

L1967

Variable y : set

L1968

Variable z : set

L1969

Variable w : set

L1970

Variable u : set

L1971

Variable v : set

L1972

Hypothesis H0 : SNo x

L1973

Hypothesis H1 : SNo y

L1974

Hypothesis H2 : SNo z

L1975

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L1976

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L1977

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L1978

Hypothesis H6 : SNo (x + y)

L1979

Hypothesis H7 : SNo ((x + y) * z)

L1980

Hypothesis H8 : SNo (x * z)

L1981

Hypothesis H9 : SNo w

L1982

Hypothesis H10 : u ∈ SNoR y

L1983

Hypothesis H11 : v ∈ SNoL z

L1984

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L1985

Hypothesis H13 : SNo u

L1986

Hypothesis H14 : y < u

L1987

Hypothesis H15 : SNo v

L1988

Hypothesis H16 : v < z

L1989

Hypothesis H17 : SNo (u * v)

L1990

Hypothesis H18 : SNo (x + u)

L1991

Hypothesis H19 : SNo (w + u * v)

L1992

Hypothesis H20 : SNo ((x + y) * v)

L1993

Hypothesis H21 : SNo (u * z)

L1994

Hypothesis H22 : SNo (x * v)

L1995

Hypothesis H24 : SNo (w + x * z)

L1996

Hypothesis H25 : SNo (u * v + x * v)

L1997

Theorem. (Conj_mul_SNo_distrR__1__23)

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

In Proofgold the corresponding term root is 03828b... and proposition id is 10109b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__2__24

L2003

Variable x : set

(*** Conj_mul_SNo_distrR__2__24 TMazBgPHPKjJzQA3vqJioPNAgeisrBBs99q bounty of about 25 bars ***)

L2004

Variable y : set

L2005

Variable z : set

L2006

Variable w : set

L2007

Variable u : set

L2008

Variable v : set

L2009

Hypothesis H0 : SNo x

L2010

Hypothesis H1 : SNo y

L2011

Hypothesis H2 : SNo z

L2012

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2013

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2014

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2015

Hypothesis H6 : SNo (x + y)

L2016

Hypothesis H7 : SNo ((x + y) * z)

L2017

Hypothesis H8 : SNo (x * z)

L2018

Hypothesis H9 : SNo w

L2019

Hypothesis H10 : u ∈ SNoR y

L2020

Hypothesis H11 : v ∈ SNoL z

L2021

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2022

Hypothesis H13 : SNo u

L2023

Hypothesis H14 : y < u

L2024

Hypothesis H15 : SNo v

L2025

Hypothesis H16 : v < z

L2026

Hypothesis H17 : SNo (u * v)

L2027

Hypothesis H18 : SNo (x + u)

L2028

Hypothesis H19 : SNo (w + u * v)

L2029

Hypothesis H20 : SNo ((x + y) * v)

L2030

Hypothesis H21 : SNo (u * z)

L2031

Hypothesis H22 : SNo (x * v)

L2032

Hypothesis H23 : SNo (y * v)

L2033

Theorem. (Conj_mul_SNo_distrR__2__24)

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

In Proofgold the corresponding term root is 34ad78... and proposition id is 6b7638...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__6__9

L2039

Variable x : set

(*** Conj_mul_SNo_distrR__6__9 TMNSZr3oLFcDZDhoLhoTNCbFXyrUmHn8Gjb bounty of about 25 bars ***)

L2040

Variable y : set

L2041

Variable z : set

L2042

Variable w : set

L2043

Variable u : set

L2044

Variable v : set

L2045

Hypothesis H0 : SNo x

L2046

Hypothesis H1 : SNo y

L2047

Hypothesis H2 : SNo z

L2048

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2049

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2050

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2051

Hypothesis H6 : SNo (x + y)

L2052

Hypothesis H7 : SNo ((x + y) * z)

L2053

Hypothesis H8 : SNo (x * z)

L2054

Hypothesis H10 : u ∈ SNoR y

L2055

Hypothesis H11 : v ∈ SNoL z

L2056

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2057

Hypothesis H13 : SNo u

L2058

Hypothesis H14 : y < u

L2059

Hypothesis H15 : SNo v

L2060

Hypothesis H16 : v < z

L2061

Hypothesis H17 : SNo (u * v)

L2062

Hypothesis H18 : SNo (x + u)

L2063

Hypothesis H19 : SNo (w + u * v)

L2064

Hypothesis H20 : SNo ((x + y) * v)

L2065

Theorem. (Conj_mul_SNo_distrR__6__9)

SNo (u * z) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 84fe3e... and proposition id is 142821...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__6__16

L2071

Variable x : set

(*** Conj_mul_SNo_distrR__6__16 TMLF1gTdLcJZPkJ1EbpYUrhXPfzyCF5hLD6 bounty of about 25 bars ***)

L2072

Variable y : set

L2073

Variable z : set

L2074

Variable w : set

L2075

Variable u : set

L2076

Variable v : set

L2077

Hypothesis H0 : SNo x

L2078

Hypothesis H1 : SNo y

L2079

Hypothesis H2 : SNo z

L2080

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2081

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2082

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2083

Hypothesis H6 : SNo (x + y)

L2084

Hypothesis H7 : SNo ((x + y) * z)

L2085

Hypothesis H8 : SNo (x * z)

L2086

Hypothesis H9 : SNo w

L2087

Hypothesis H10 : u ∈ SNoR y

L2088

Hypothesis H11 : v ∈ SNoL z

L2089

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2090

Hypothesis H13 : SNo u

L2091

Hypothesis H14 : y < u

L2092

Hypothesis H15 : SNo v

L2093

Hypothesis H17 : SNo (u * v)

L2094

Hypothesis H18 : SNo (x + u)

L2095

Hypothesis H19 : SNo (w + u * v)

L2096

Hypothesis H20 : SNo ((x + y) * v)

L2097

Theorem. (Conj_mul_SNo_distrR__6__16)

SNo (u * z) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 3156cd... and proposition id is 16c789...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__6__19

L2103

Variable x : set

(*** Conj_mul_SNo_distrR__6__19 TMdxxHpALSLaGmxwazURBagYrm8Wg6Du7Yb bounty of about 25 bars ***)

L2104

Variable y : set

L2105

Variable z : set

L2106

Variable w : set

L2107

Variable u : set

L2108

Variable v : set

L2109

Hypothesis H0 : SNo x

L2110

Hypothesis H1 : SNo y

L2111

Hypothesis H2 : SNo z

L2112

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2113

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2114

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2115

Hypothesis H6 : SNo (x + y)

L2116

Hypothesis H7 : SNo ((x + y) * z)

L2117

Hypothesis H8 : SNo (x * z)

L2118

Hypothesis H9 : SNo w

L2119

Hypothesis H10 : u ∈ SNoR y

L2120

Hypothesis H11 : v ∈ SNoL z

L2121

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2122

Hypothesis H13 : SNo u

L2123

Hypothesis H14 : y < u

L2124

Hypothesis H15 : SNo v

L2125

Hypothesis H16 : v < z

L2126

Hypothesis H17 : SNo (u * v)

L2127

Hypothesis H18 : SNo (x + u)

L2128

Hypothesis H20 : SNo ((x + y) * v)

L2129

Theorem. (Conj_mul_SNo_distrR__6__19)

SNo (u * z) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 00583d... and proposition id is 08638b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__6__20

L2135

Variable x : set

(*** Conj_mul_SNo_distrR__6__20 TMLLNWir96k4Kmo3V5RG9qHaPs6p4VpTM94 bounty of about 25 bars ***)

L2136

Variable y : set

L2137

Variable z : set

L2138

Variable w : set

L2139

Variable u : set

L2140

Variable v : set

L2141

Hypothesis H0 : SNo x

L2142

Hypothesis H1 : SNo y

L2143

Hypothesis H2 : SNo z

L2144

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2145

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2146

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2147

Hypothesis H6 : SNo (x + y)

L2148

Hypothesis H7 : SNo ((x + y) * z)

L2149

Hypothesis H8 : SNo (x * z)

L2150

Hypothesis H9 : SNo w

L2151

Hypothesis H10 : u ∈ SNoR y

L2152

Hypothesis H11 : v ∈ SNoL z

L2153

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2154

Hypothesis H13 : SNo u

L2155

Hypothesis H14 : y < u

L2156

Hypothesis H15 : SNo v

L2157

Hypothesis H16 : v < z

L2158

Hypothesis H17 : SNo (u * v)

L2159

Hypothesis H18 : SNo (x + u)

L2160

Hypothesis H19 : SNo (w + u * v)

L2161

Theorem. (Conj_mul_SNo_distrR__6__20)

SNo (u * z) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 0ed748... and proposition id is 50dcb6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__10__4

L2167

Variable x : set

(*** Conj_mul_SNo_distrR__10__4 TMVSjGET2zA9sdLvsn5y86PffrT8JuvbfDN bounty of about 25 bars ***)

L2168

Variable y : set

L2169

Variable z : set

L2170

Variable w : set

L2171

Variable u : set

L2172

Variable v : set

L2173

Hypothesis H0 : SNo x

L2174

Hypothesis H1 : SNo y

L2175

Hypothesis H2 : SNo z

L2176

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2177

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2178

Hypothesis H6 : SNo (x + y)

L2179

Hypothesis H7 : SNo ((x + y) * z)

L2180

Hypothesis H8 : SNo (x * z)

L2181

Hypothesis H9 : SNo w

L2182

Hypothesis H10 : u ∈ SNoR y

L2183

Hypothesis H11 : v ∈ SNoL z

L2184

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2185

Hypothesis H13 : SNo u

L2186

Hypothesis H14 : y < u

L2187

Hypothesis H15 : SNo v

L2188

Hypothesis H16 : v < z

L2189

Theorem. (Conj_mul_SNo_distrR__10__4)

SNo (u * v) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 433a64... and proposition id is 3481c1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__13__1

L2195

Variable x : set

(*** Conj_mul_SNo_distrR__13__1 TMTEhWgawfBdy8XFKt6VMfc7apPZs2QEu7h bounty of about 25 bars ***)

L2196

Variable y : set

L2197

Variable z : set

L2198

Variable w : set

L2199

Variable u : set

L2200

Variable v : set

L2201

Hypothesis H0 : SNo x

L2202

Hypothesis H2 : SNo z

L2203

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2204

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2205

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2206

Hypothesis H6 : SNo (x + y)

L2207

Hypothesis H7 : SNo ((x + y) * z)

L2208

Hypothesis H8 : SNo (x * z)

L2209

Hypothesis H9 : SNo w

L2210

Hypothesis H10 : u ∈ SNoL y

L2211

Hypothesis H11 : v ∈ SNoR z

L2212

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2213

Hypothesis H13 : SNo u

L2214

Hypothesis H14 : u < y

L2215

Hypothesis H15 : SNo v

L2216

Hypothesis H16 : z < v

L2217

Hypothesis H17 : SNo (u * v)

L2218

Hypothesis H18 : SNo (x + u)

L2219

Hypothesis H19 : SNo (w + u * v)

L2220

Hypothesis H20 : SNo ((x + y) * v)

L2221

Hypothesis H21 : SNo (u * z)

L2222

Hypothesis H22 : SNo (x * v)

L2223

Theorem. (Conj_mul_SNo_distrR__13__1)

SNo (y * v) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is bc6909... and proposition id is 8baa42...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__14__14

L2229

Variable x : set

(*** Conj_mul_SNo_distrR__14__14 TMUq3koYdevu92BHQdKyTBrY49UCyUowryc bounty of about 25 bars ***)

L2230

Variable y : set

L2231

Variable z : set

L2232

Variable w : set

L2233

Variable u : set

L2234

Variable v : set

L2235

Hypothesis H0 : SNo x

L2236

Hypothesis H1 : SNo y

L2237

Hypothesis H2 : SNo z

L2238

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2239

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2240

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2241

Hypothesis H6 : SNo (x + y)

L2242

Hypothesis H7 : SNo ((x + y) * z)

L2243

Hypothesis H8 : SNo (x * z)

L2244

Hypothesis H9 : SNo w

L2245

Hypothesis H10 : u ∈ SNoL y

L2246

Hypothesis H11 : v ∈ SNoR z

L2247

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2248

Hypothesis H13 : SNo u

L2249

Hypothesis H15 : SNo v

L2250

Hypothesis H16 : z < v

L2251

Hypothesis H17 : SNo (u * v)

L2252

Hypothesis H18 : SNo (x + u)

L2253

Hypothesis H19 : SNo (w + u * v)

L2254

Hypothesis H20 : SNo ((x + y) * v)

L2255

Hypothesis H21 : SNo (u * z)

L2256

Theorem. (Conj_mul_SNo_distrR__14__14)

SNo (x * v) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 458525... and proposition id is 1b699a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__15__1

L2262

Variable x : set

(*** Conj_mul_SNo_distrR__15__1 TMGGJsJaGiRqTZ6wifDKMkdsnqS3WZbin1Q bounty of about 25 bars ***)

L2263

Variable y : set

L2264

Variable z : set

L2265

Variable w : set

L2266

Variable u : set

L2267

Variable v : set

L2268

Hypothesis H0 : SNo x

L2269

Hypothesis H2 : SNo z

L2270

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2271

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2272

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2273

Hypothesis H6 : SNo (x + y)

L2274

Hypothesis H7 : SNo ((x + y) * z)

L2275

Hypothesis H8 : SNo (x * z)

L2276

Hypothesis H9 : SNo w

L2277

Hypothesis H10 : u ∈ SNoL y

L2278

Hypothesis H11 : v ∈ SNoR z

L2279

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2280

Hypothesis H13 : SNo u

L2281

Hypothesis H14 : u < y

L2282

Hypothesis H15 : SNo v

L2283

Hypothesis H16 : z < v

L2284

Hypothesis H17 : SNo (u * v)

L2285

Hypothesis H18 : SNo (x + u)

L2286

Hypothesis H19 : SNo (w + u * v)

L2287

Hypothesis H20 : SNo ((x + y) * v)

L2288

Theorem. (Conj_mul_SNo_distrR__15__1)

SNo (u * z) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 9168e9... and proposition id is 38d590...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__16__5

L2294

Variable x : set

(*** Conj_mul_SNo_distrR__16__5 TMXFv64KLyKReDZqYSUDqEygAh9yGBYAes4 bounty of about 25 bars ***)

L2295

Variable y : set

L2296

Variable z : set

L2297

Variable w : set

L2298

Variable u : set

L2299

Variable v : set

L2300

Hypothesis H0 : SNo x

L2301

Hypothesis H1 : SNo y

L2302

Hypothesis H2 : SNo z

L2303

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2304

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2305

Hypothesis H6 : SNo (x + y)

L2306

Hypothesis H7 : SNo ((x + y) * z)

L2307

Hypothesis H8 : SNo (x * z)

L2308

Hypothesis H9 : SNo w

L2309

Hypothesis H10 : u ∈ SNoL y

L2310

Hypothesis H11 : v ∈ SNoR z

L2311

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2312

Hypothesis H13 : SNo u

L2313

Hypothesis H14 : u < y

L2314

Hypothesis H15 : SNo v

L2315

Hypothesis H16 : z < v

L2316

Hypothesis H17 : SNo (u * v)

L2317

Hypothesis H18 : SNo (x + u)

L2318

Hypothesis H19 : SNo (w + u * v)

L2319

Theorem. (Conj_mul_SNo_distrR__16__5)

SNo ((x + y) * v) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 1dd9b6... and proposition id is d0cfdd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__16__14

L2325

Variable x : set

(*** Conj_mul_SNo_distrR__16__14 TMbHYN5Jyo2XmG5ayyk39C3mLQ5yx4SnMrt bounty of about 25 bars ***)

L2326

Variable y : set

L2327

Variable z : set

L2328

Variable w : set

L2329

Variable u : set

L2330

Variable v : set

L2331

Hypothesis H0 : SNo x

L2332

Hypothesis H1 : SNo y

L2333

Hypothesis H2 : SNo z

L2334

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2335

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2336

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2337

Hypothesis H6 : SNo (x + y)

L2338

Hypothesis H7 : SNo ((x + y) * z)

L2339

Hypothesis H8 : SNo (x * z)

L2340

Hypothesis H9 : SNo w

L2341

Hypothesis H10 : u ∈ SNoL y

L2342

Hypothesis H11 : v ∈ SNoR z

L2343

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2344

Hypothesis H13 : SNo u

L2345

Hypothesis H15 : SNo v

L2346

Hypothesis H16 : z < v

L2347

Hypothesis H17 : SNo (u * v)

L2348

Hypothesis H18 : SNo (x + u)

L2349

Hypothesis H19 : SNo (w + u * v)

L2350

Theorem. (Conj_mul_SNo_distrR__16__14)

SNo ((x + y) * v) → (x + y) * z < w + x * z

In Proofgold the corresponding term root is 944ff3... and proposition id is f96a21...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__17__1

L2356

Variable x : set

(*** Conj_mul_SNo_distrR__17__1 TMPuETwoonChvvNss6d9kC6fwcTSjz7ypHe bounty of about 25 bars ***)

L2357

Variable y : set

L2358

Variable z : set

L2359

Variable w : set

L2360

Variable u : set

L2361

Variable v : set

L2362

Hypothesis H0 : SNo x

L2363

Hypothesis H2 : SNo z

L2364

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2365

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2366

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2367

Hypothesis H6 : SNo (x + y)

L2368

Hypothesis H7 : SNo ((x + y) * z)

L2369

Hypothesis H8 : SNo (x * z)

L2370

Hypothesis H9 : SNo w

L2371

Hypothesis H10 : u ∈ SNoL y

L2372

Hypothesis H11 : v ∈ SNoR z

L2373

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2374

Hypothesis H13 : SNo u

L2375

Hypothesis H14 : u < y

L2376

Hypothesis H15 : SNo v

L2377

Hypothesis H16 : z < v

L2378

Hypothesis H17 : SNo (u * v)

L2379

Hypothesis H18 : SNo (x + u)

L2380

Theorem. (Conj_mul_SNo_distrR__17__1)

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

In Proofgold the corresponding term root is 656b77... and proposition id is 021642...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__17__2

L2386

Variable x : set

(*** Conj_mul_SNo_distrR__17__2 TMdLKZsNUGogTgW6KRAyLXttmcRqbKHwvE7 bounty of about 25 bars ***)

L2387

Variable y : set

L2388

Variable z : set

L2389

Variable w : set

L2390

Variable u : set

L2391

Variable v : set

L2392

Hypothesis H0 : SNo x

L2393

Hypothesis H1 : SNo y

L2394

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2395

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2396

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2397

Hypothesis H6 : SNo (x + y)

L2398

Hypothesis H7 : SNo ((x + y) * z)

L2399

Hypothesis H8 : SNo (x * z)

L2400

Hypothesis H9 : SNo w

L2401

Hypothesis H10 : u ∈ SNoL y

L2402

Hypothesis H11 : v ∈ SNoR z

L2403

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2404

Hypothesis H13 : SNo u

L2405

Hypothesis H14 : u < y

L2406

Hypothesis H15 : SNo v

L2407

Hypothesis H16 : z < v

L2408

Hypothesis H17 : SNo (u * v)

L2409

Hypothesis H18 : SNo (x + u)

L2410

Theorem. (Conj_mul_SNo_distrR__17__2)

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

In Proofgold the corresponding term root is f88ba5... and proposition id is 2bfe2e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__17__11

L2416

Variable x : set

(*** Conj_mul_SNo_distrR__17__11 TMVze8iNfeViJfn7XUDSgPbCFnLZoiPXyzt bounty of about 25 bars ***)

L2417

Variable y : set

L2418

Variable z : set

L2419

Variable w : set

L2420

Variable u : set

L2421

Variable v : set

L2422

Hypothesis H0 : SNo x

L2423

Hypothesis H1 : SNo y

L2424

Hypothesis H2 : SNo z

L2425

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L2426

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2427

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L2428

Hypothesis H6 : SNo (x + y)

L2429

Hypothesis H7 : SNo ((x + y) * z)

L2430

Hypothesis H8 : SNo (x * z)

L2431

Hypothesis H9 : SNo w

L2432

Hypothesis H10 : u ∈ SNoL y

L2433

Hypothesis H12 : (u * z + y * v) ≤ w + u * v

L2434

Hypothesis H13 : SNo u

L2435

Hypothesis H14 : u < y

L2436

Hypothesis H15 : SNo v

L2437

Hypothesis H16 : z < v

L2438

Hypothesis H17 : SNo (u * v)

L2439

Hypothesis H18 : SNo (x + u)

L2440

Theorem. (Conj_mul_SNo_distrR__17__11)

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

In Proofgold the corresponding term root is 12d30e... and proposition id is 4f9207...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__20__1

L2446

Variable x : set

(*** Conj_mul_SNo_distrR__20__1 TMdYru46QWaaihpFFYaDqdod8143ZaBkFhY bounty of about 25 bars ***)

L2447

Variable y : set

L2448

Variable z : set

L2449

Variable w : set

L2450

Variable u : set

L2451

Variable v : set

L2452

Hypothesis H0 : SNo x

L2453

Hypothesis H2 : SNo z

L2454

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2455

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2456

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2457

Hypothesis H6 : SNo (x + y)

L2458

Hypothesis H7 : SNo ((x + y) * z)

L2459

Hypothesis H8 : SNo (y * z)

L2460

Hypothesis H9 : SNo w

L2461

Hypothesis H10 : u ∈ SNoR x

L2462

Hypothesis H11 : v ∈ SNoL z

L2463

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2464

Hypothesis H13 : SNo u

L2465

Hypothesis H14 : x < u

L2466

Hypothesis H15 : SNo v

L2467

Hypothesis H16 : v < z

L2468

Hypothesis H17 : SNo (u * v)

L2469

Hypothesis H18 : SNo (u + y)

L2470

Hypothesis H19 : SNo (w + u * v)

L2471

Hypothesis H20 : SNo ((x + y) * v)

L2472

Hypothesis H21 : SNo (u * z)

L2473

Hypothesis H22 : SNo (x * v)

L2474

Hypothesis H23 : SNo (y * v)

L2475

Hypothesis H24 : SNo (w + y * z)

L2476

Theorem. (Conj_mul_SNo_distrR__20__1)

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

In Proofgold the corresponding term root is dd2be2... and proposition id is c8fd98...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__20__2

L2482

Variable x : set

(*** Conj_mul_SNo_distrR__20__2 TMLRWDXYDhEao3t4zgcFWBKsNTmeQUPTCYp bounty of about 25 bars ***)

L2483

Variable y : set

L2484

Variable z : set

L2485

Variable w : set

L2486

Variable u : set

L2487

Variable v : set

L2488

Hypothesis H0 : SNo x

L2489

Hypothesis H1 : SNo y

L2490

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2491

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2492

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2493

Hypothesis H6 : SNo (x + y)

L2494

Hypothesis H7 : SNo ((x + y) * z)

L2495

Hypothesis H8 : SNo (y * z)

L2496

Hypothesis H9 : SNo w

L2497

Hypothesis H10 : u ∈ SNoR x

L2498

Hypothesis H11 : v ∈ SNoL z

L2499

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2500

Hypothesis H13 : SNo u

L2501

Hypothesis H14 : x < u

L2502

Hypothesis H15 : SNo v

L2503

Hypothesis H16 : v < z

L2504

Hypothesis H17 : SNo (u * v)

L2505

Hypothesis H18 : SNo (u + y)

L2506

Hypothesis H19 : SNo (w + u * v)

L2507

Hypothesis H20 : SNo ((x + y) * v)

L2508

Hypothesis H21 : SNo (u * z)

L2509

Hypothesis H22 : SNo (x * v)

L2510

Hypothesis H23 : SNo (y * v)

L2511

Hypothesis H24 : SNo (w + y * z)

L2512

Theorem. (Conj_mul_SNo_distrR__20__2)

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

In Proofgold the corresponding term root is 97bede... and proposition id is a465f9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__20__6

L2518

Variable x : set

(*** Conj_mul_SNo_distrR__20__6 TMXQrkec7a2mNMJsBS6wbQz7s35Y21L1PUv bounty of about 25 bars ***)

L2519

Variable y : set

L2520

Variable z : set

L2521

Variable w : set

L2522

Variable u : set

L2523

Variable v : set

L2524

Hypothesis H0 : SNo x

L2525

Hypothesis H1 : SNo y

L2526

Hypothesis H2 : SNo z

L2527

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2528

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2529

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2530

Hypothesis H7 : SNo ((x + y) * z)

L2531

Hypothesis H8 : SNo (y * z)

L2532

Hypothesis H9 : SNo w

L2533

Hypothesis H10 : u ∈ SNoR x

L2534

Hypothesis H11 : v ∈ SNoL z

L2535

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2536

Hypothesis H13 : SNo u

L2537

Hypothesis H14 : x < u

L2538

Hypothesis H15 : SNo v

L2539

Hypothesis H16 : v < z

L2540

Hypothesis H17 : SNo (u * v)

L2541

Hypothesis H18 : SNo (u + y)

L2542

Hypothesis H19 : SNo (w + u * v)

L2543

Hypothesis H20 : SNo ((x + y) * v)

L2544

Hypothesis H21 : SNo (u * z)

L2545

Hypothesis H22 : SNo (x * v)

L2546

Hypothesis H23 : SNo (y * v)

L2547

Hypothesis H24 : SNo (w + y * z)

L2548

Theorem. (Conj_mul_SNo_distrR__20__6)

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

In Proofgold the corresponding term root is 8268de... and proposition id is 07d2e6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__20__20

L2554

Variable x : set

(*** Conj_mul_SNo_distrR__20__20 TMTgvx3t4CMs73L8K3uzWULzs6SS4HYMWCy bounty of about 25 bars ***)

L2555

Variable y : set

L2556

Variable z : set

L2557

Variable w : set

L2558

Variable u : set

L2559

Variable v : set

L2560

Hypothesis H0 : SNo x

L2561

Hypothesis H1 : SNo y

L2562

Hypothesis H2 : SNo z

L2563

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2564

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2565

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2566

Hypothesis H6 : SNo (x + y)

L2567

Hypothesis H7 : SNo ((x + y) * z)

L2568

Hypothesis H8 : SNo (y * z)

L2569

Hypothesis H9 : SNo w

L2570

Hypothesis H10 : u ∈ SNoR x

L2571

Hypothesis H11 : v ∈ SNoL z

L2572

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2573

Hypothesis H13 : SNo u

L2574

Hypothesis H14 : x < u

L2575

Hypothesis H15 : SNo v

L2576

Hypothesis H16 : v < z

L2577

Hypothesis H17 : SNo (u * v)

L2578

Hypothesis H18 : SNo (u + y)

L2579

Hypothesis H19 : SNo (w + u * v)

L2580

Hypothesis H21 : SNo (u * z)

L2581

Hypothesis H22 : SNo (x * v)

L2582

Hypothesis H23 : SNo (y * v)

L2583

Hypothesis H24 : SNo (w + y * z)

L2584

Theorem. (Conj_mul_SNo_distrR__20__20)

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

In Proofgold the corresponding term root is c3f9cf... and proposition id is 384e93...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__25__5

L2590

Variable x : set

(*** Conj_mul_SNo_distrR__25__5 TMHjsjSLG3qcx6wNuMp7VRxUdWLqpmYjoYV bounty of about 25 bars ***)

L2591

Variable y : set

L2592

Variable z : set

L2593

Variable w : set

L2594

Variable u : set

L2595

Variable v : set

L2596

Hypothesis H0 : SNo x

L2597

Hypothesis H1 : SNo y

L2598

Hypothesis H2 : SNo z

L2599

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2600

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2601

Hypothesis H6 : SNo (x + y)

L2602

Hypothesis H7 : SNo ((x + y) * z)

L2603

Hypothesis H8 : SNo (y * z)

L2604

Hypothesis H9 : SNo w

L2605

Hypothesis H10 : u ∈ SNoR x

L2606

Hypothesis H11 : v ∈ SNoL z

L2607

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2608

Hypothesis H13 : SNo u

L2609

Hypothesis H14 : x < u

L2610

Hypothesis H15 : SNo v

L2611

Hypothesis H16 : v < z

L2612

Hypothesis H17 : SNo (u * v)

L2613

Hypothesis H18 : SNo (u + y)

L2614

Hypothesis H19 : SNo (w + u * v)

L2615

Theorem. (Conj_mul_SNo_distrR__25__5)

SNo ((x + y) * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 741a11... and proposition id is a26c43...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__27__13

L2621

Variable x : set

(*** Conj_mul_SNo_distrR__27__13 TMJBVG5EHRU5e6vemxeuy2J8D5bHuxYobns bounty of about 25 bars ***)

L2622

Variable y : set

L2623

Variable z : set

L2624

Variable w : set

L2625

Variable u : set

L2626

Variable v : set

L2627

Hypothesis H0 : SNo x

L2628

Hypothesis H1 : SNo y

L2629

Hypothesis H2 : SNo z

L2630

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2631

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2632

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2633

Hypothesis H6 : SNo (x + y)

L2634

Hypothesis H7 : SNo ((x + y) * z)

L2635

Hypothesis H8 : SNo (y * z)

L2636

Hypothesis H9 : SNo w

L2637

Hypothesis H10 : u ∈ SNoR x

L2638

Hypothesis H11 : v ∈ SNoL z

L2639

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2640

Hypothesis H14 : x < u

L2641

Hypothesis H15 : SNo v

L2642

Hypothesis H16 : v < z

L2643

Hypothesis H17 : SNo (u * v)

L2644

Theorem. (Conj_mul_SNo_distrR__27__13)

SNo (u + y) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 531103... and proposition id is e6b186...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__27__16

L2650

Variable x : set

(*** Conj_mul_SNo_distrR__27__16 TMb3G8kqeGB5A4xTUwNvQtAsyhpDPi3Z7Va bounty of about 25 bars ***)

L2651

Variable y : set

L2652

Variable z : set

L2653

Variable w : set

L2654

Variable u : set

L2655

Variable v : set

L2656

Hypothesis H0 : SNo x

L2657

Hypothesis H1 : SNo y

L2658

Hypothesis H2 : SNo z

L2659

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2660

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2661

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2662

Hypothesis H6 : SNo (x + y)

L2663

Hypothesis H7 : SNo ((x + y) * z)

L2664

Hypothesis H8 : SNo (y * z)

L2665

Hypothesis H9 : SNo w

L2666

Hypothesis H10 : u ∈ SNoR x

L2667

Hypothesis H11 : v ∈ SNoL z

L2668

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2669

Hypothesis H13 : SNo u

L2670

Hypothesis H14 : x < u

L2671

Hypothesis H15 : SNo v

L2672

Hypothesis H17 : SNo (u * v)

L2673

Theorem. (Conj_mul_SNo_distrR__27__16)

SNo (u + y) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 6b997a... and proposition id is 12b733...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__29__11

L2679

Variable x : set

(*** Conj_mul_SNo_distrR__29__11 TMQsqQRVLYSr2WFnLC5ASCoxyckBKaMzWQC bounty of about 25 bars ***)

L2680

Variable y : set

L2681

Variable z : set

L2682

Variable w : set

L2683

Variable u : set

L2684

Variable v : set

L2685

Hypothesis H0 : SNo x

L2686

Hypothesis H1 : SNo y

L2687

Hypothesis H2 : SNo z

L2688

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2689

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2690

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2691

Hypothesis H6 : SNo (x + y)

L2692

Hypothesis H7 : SNo ((x + y) * z)

L2693

Hypothesis H8 : SNo (y * z)

L2694

Hypothesis H9 : SNo w

L2695

Hypothesis H10 : u ∈ SNoL x

L2696

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2697

Hypothesis H13 : SNo u

L2698

Hypothesis H14 : u < x

L2699

Hypothesis H15 : SNo v

L2700

Hypothesis H16 : z < v

L2701

Hypothesis H17 : SNo (u * v)

L2702

Hypothesis H18 : SNo (u + y)

L2703

Hypothesis H19 : SNo (w + u * v)

L2704

Hypothesis H20 : SNo ((x + y) * v)

L2705

Hypothesis H21 : SNo (u * z)

L2706

Hypothesis H22 : SNo (x * v)

L2707

Hypothesis H23 : SNo (y * v)

L2708

Hypothesis H24 : SNo (w + y * z)

L2709

Hypothesis H25 : SNo (u * v + y * v)

L2710

Theorem. (Conj_mul_SNo_distrR__29__11)

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

In Proofgold the corresponding term root is d5196f... and proposition id is ab2580...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__30__18

L2716

Variable x : set

(*** Conj_mul_SNo_distrR__30__18 TMK2jdeA3NkuqpWLuEZFmRPtpKycPqS79mL bounty of about 25 bars ***)

L2717

Variable y : set

L2718

Variable z : set

L2719

Variable w : set

L2720

Variable u : set

L2721

Variable v : set

L2722

Hypothesis H0 : SNo x

L2723

Hypothesis H1 : SNo y

L2724

Hypothesis H2 : SNo z

L2725

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2726

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2727

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2728

Hypothesis H6 : SNo (x + y)

L2729

Hypothesis H7 : SNo ((x + y) * z)

L2730

Hypothesis H8 : SNo (y * z)

L2731

Hypothesis H9 : SNo w

L2732

Hypothesis H10 : u ∈ SNoL x

L2733

Hypothesis H11 : v ∈ SNoR z

L2734

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2735

Hypothesis H13 : SNo u

L2736

Hypothesis H14 : u < x

L2737

Hypothesis H15 : SNo v

L2738

Hypothesis H16 : z < v

L2739

Hypothesis H17 : SNo (u * v)

L2740

Hypothesis H19 : SNo (w + u * v)

L2741

Hypothesis H20 : SNo ((x + y) * v)

L2742

Hypothesis H21 : SNo (u * z)

L2743

Hypothesis H22 : SNo (x * v)

L2744

Hypothesis H23 : SNo (y * v)

L2745

Hypothesis H24 : SNo (w + y * z)

L2746

Theorem. (Conj_mul_SNo_distrR__30__18)

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

In Proofgold the corresponding term root is 42313c... and proposition id is c9bfbf...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__31__15

L2752

Variable x : set

(*** Conj_mul_SNo_distrR__31__15 TMLPm7nvQmkx7oyDa8wbSdqZToJpQaT2VUo bounty of about 25 bars ***)

L2753

Variable y : set

L2754

Variable z : set

L2755

Variable w : set

L2756

Variable u : set

L2757

Variable v : set

L2758

Hypothesis H0 : SNo x

L2759

Hypothesis H1 : SNo y

L2760

Hypothesis H2 : SNo z

L2761

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2762

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2763

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2764

Hypothesis H6 : SNo (x + y)

L2765

Hypothesis H7 : SNo ((x + y) * z)

L2766

Hypothesis H8 : SNo (y * z)

L2767

Hypothesis H9 : SNo w

L2768

Hypothesis H10 : u ∈ SNoL x

L2769

Hypothesis H11 : v ∈ SNoR z

L2770

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2771

Hypothesis H13 : SNo u

L2772

Hypothesis H14 : u < x

L2773

Hypothesis H16 : z < v

L2774

Hypothesis H17 : SNo (u * v)

L2775

Hypothesis H18 : SNo (u + y)

L2776

Hypothesis H19 : SNo (w + u * v)

L2777

Hypothesis H20 : SNo ((x + y) * v)

L2778

Hypothesis H21 : SNo (u * z)

L2779

Hypothesis H22 : SNo (x * v)

L2780

Hypothesis H23 : SNo (y * v)

L2781

Theorem. (Conj_mul_SNo_distrR__31__15)

SNo (w + y * z) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 8963bb... and proposition id is 683dd8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__31__19

L2787

Variable x : set

(*** Conj_mul_SNo_distrR__31__19 TMUNHnKt24Q6oTQFXL3RDHCupi4sJZD1MwG bounty of about 25 bars ***)

L2788

Variable y : set

L2789

Variable z : set

L2790

Variable w : set

L2791

Variable u : set

L2792

Variable v : set

L2793

Hypothesis H0 : SNo x

L2794

Hypothesis H1 : SNo y

L2795

Hypothesis H2 : SNo z

L2796

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2797

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2798

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2799

Hypothesis H6 : SNo (x + y)

L2800

Hypothesis H7 : SNo ((x + y) * z)

L2801

Hypothesis H8 : SNo (y * z)

L2802

Hypothesis H9 : SNo w

L2803

Hypothesis H10 : u ∈ SNoL x

L2804

Hypothesis H11 : v ∈ SNoR z

L2805

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2806

Hypothesis H13 : SNo u

L2807

Hypothesis H14 : u < x

L2808

Hypothesis H15 : SNo v

L2809

Hypothesis H16 : z < v

L2810

Hypothesis H17 : SNo (u * v)

L2811

Hypothesis H18 : SNo (u + y)

L2812

Hypothesis H20 : SNo ((x + y) * v)

L2813

Hypothesis H21 : SNo (u * z)

L2814

Hypothesis H22 : SNo (x * v)

L2815

Hypothesis H23 : SNo (y * v)

L2816

Theorem. (Conj_mul_SNo_distrR__31__19)

SNo (w + y * z) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is a5f24c... and proposition id is f61bc4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__32__11

L2822

Variable x : set

(*** Conj_mul_SNo_distrR__32__11 TMNRNpskpRrbKUGfXvaTQsZKjyqsxogtBuP bounty of about 25 bars ***)

L2823

Variable y : set

L2824

Variable z : set

L2825

Variable w : set

L2826

Variable u : set

L2827

Variable v : set

L2828

Hypothesis H0 : SNo x

L2829

Hypothesis H1 : SNo y

L2830

Hypothesis H2 : SNo z

L2831

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2832

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2833

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2834

Hypothesis H6 : SNo (x + y)

L2835

Hypothesis H7 : SNo ((x + y) * z)

L2836

Hypothesis H8 : SNo (y * z)

L2837

Hypothesis H9 : SNo w

L2838

Hypothesis H10 : u ∈ SNoL x

L2839

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2840

Hypothesis H13 : SNo u

L2841

Hypothesis H14 : u < x

L2842

Hypothesis H15 : SNo v

L2843

Hypothesis H16 : z < v

L2844

Hypothesis H17 : SNo (u * v)

L2845

Hypothesis H18 : SNo (u + y)

L2846

Hypothesis H19 : SNo (w + u * v)

L2847

Hypothesis H20 : SNo ((x + y) * v)

L2848

Hypothesis H21 : SNo (u * z)

L2849

Hypothesis H22 : SNo (x * v)

L2850

Theorem. (Conj_mul_SNo_distrR__32__11)

SNo (y * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is e2c99e... and proposition id is 8adcdd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__32__14

L2856

Variable x : set

(*** Conj_mul_SNo_distrR__32__14 TMRfuofqd6iKhka1dWPu3pU7P1bkiE3pQNL bounty of about 25 bars ***)

L2857

Variable y : set

L2858

Variable z : set

L2859

Variable w : set

L2860

Variable u : set

L2861

Variable v : set

L2862

Hypothesis H0 : SNo x

L2863

Hypothesis H1 : SNo y

L2864

Hypothesis H2 : SNo z

L2865

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2866

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2867

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2868

Hypothesis H6 : SNo (x + y)

L2869

Hypothesis H7 : SNo ((x + y) * z)

L2870

Hypothesis H8 : SNo (y * z)

L2871

Hypothesis H9 : SNo w

L2872

Hypothesis H10 : u ∈ SNoL x

L2873

Hypothesis H11 : v ∈ SNoR z

L2874

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2875

Hypothesis H13 : SNo u

L2876

Hypothesis H15 : SNo v

L2877

Hypothesis H16 : z < v

L2878

Hypothesis H17 : SNo (u * v)

L2879

Hypothesis H18 : SNo (u + y)

L2880

Hypothesis H19 : SNo (w + u * v)

L2881

Hypothesis H20 : SNo ((x + y) * v)

L2882

Hypothesis H21 : SNo (u * z)

L2883

Hypothesis H22 : SNo (x * v)

L2884

Theorem. (Conj_mul_SNo_distrR__32__14)

SNo (y * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 74e4eb... and proposition id is 67bfd0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__32__16

L2890

Variable x : set

(*** Conj_mul_SNo_distrR__32__16 TMReDWJ14kaVXeicSxzY37Tv1r4vQ4VERa7 bounty of about 25 bars ***)

L2891

Variable y : set

L2892

Variable z : set

L2893

Variable w : set

L2894

Variable u : set

L2895

Variable v : set

L2896

Hypothesis H0 : SNo x

L2897

Hypothesis H1 : SNo y

L2898

Hypothesis H2 : SNo z

L2899

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2900

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2901

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2902

Hypothesis H6 : SNo (x + y)

L2903

Hypothesis H7 : SNo ((x + y) * z)

L2904

Hypothesis H8 : SNo (y * z)

L2905

Hypothesis H9 : SNo w

L2906

Hypothesis H10 : u ∈ SNoL x

L2907

Hypothesis H11 : v ∈ SNoR z

L2908

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2909

Hypothesis H13 : SNo u

L2910

Hypothesis H14 : u < x

L2911

Hypothesis H15 : SNo v

L2912

Hypothesis H17 : SNo (u * v)

L2913

Hypothesis H18 : SNo (u + y)

L2914

Hypothesis H19 : SNo (w + u * v)

L2915

Hypothesis H20 : SNo ((x + y) * v)

L2916

Hypothesis H21 : SNo (u * z)

L2917

Hypothesis H22 : SNo (x * v)

L2918

Theorem. (Conj_mul_SNo_distrR__32__16)

SNo (y * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is dd084b... and proposition id is 04aa8e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__32__22

L2924

Variable x : set

(*** Conj_mul_SNo_distrR__32__22 TMKqvFAR6LgY8ChXr3EykqYgMKehVFoSAG4 bounty of about 25 bars ***)

L2925

Variable y : set

L2926

Variable z : set

L2927

Variable w : set

L2928

Variable u : set

L2929

Variable v : set

L2930

Hypothesis H0 : SNo x

L2931

Hypothesis H1 : SNo y

L2932

Hypothesis H2 : SNo z

L2933

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2934

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2935

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2936

Hypothesis H6 : SNo (x + y)

L2937

Hypothesis H7 : SNo ((x + y) * z)

L2938

Hypothesis H8 : SNo (y * z)

L2939

Hypothesis H9 : SNo w

L2940

Hypothesis H10 : u ∈ SNoL x

L2941

Hypothesis H11 : v ∈ SNoR z

L2942

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2943

Hypothesis H13 : SNo u

L2944

Hypothesis H14 : u < x

L2945

Hypothesis H15 : SNo v

L2946

Hypothesis H16 : z < v

L2947

Hypothesis H17 : SNo (u * v)

L2948

Hypothesis H18 : SNo (u + y)

L2949

Hypothesis H19 : SNo (w + u * v)

L2950

Hypothesis H20 : SNo ((x + y) * v)

L2951

Hypothesis H21 : SNo (u * z)

L2952

Theorem. (Conj_mul_SNo_distrR__32__22)

SNo (y * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 20f11d... and proposition id is e19799...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__33__1

L2958

Variable x : set

(*** Conj_mul_SNo_distrR__33__1 TMYCzbanTRopFqWvkPR1SC6UgRwXDcAv3WX bounty of about 25 bars ***)

L2959

Variable y : set

L2960

Variable z : set

L2961

Variable w : set

L2962

Variable u : set

L2963

Variable v : set

L2964

Hypothesis H0 : SNo x

L2965

Hypothesis H2 : SNo z

L2966

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L2967

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L2968

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L2969

Hypothesis H6 : SNo (x + y)

L2970

Hypothesis H7 : SNo ((x + y) * z)

L2971

Hypothesis H8 : SNo (y * z)

L2972

Hypothesis H9 : SNo w

L2973

Hypothesis H10 : u ∈ SNoL x

L2974

Hypothesis H11 : v ∈ SNoR z

L2975

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L2976

Hypothesis H13 : SNo u

L2977

Hypothesis H14 : u < x

L2978

Hypothesis H15 : SNo v

L2979

Hypothesis H16 : z < v

L2980

Hypothesis H17 : SNo (u * v)

L2981

Hypothesis H18 : SNo (u + y)

L2982

Hypothesis H19 : SNo (w + u * v)

L2983

Hypothesis H20 : SNo ((x + y) * v)

L2984

Hypothesis H21 : SNo (u * z)

L2985

Theorem. (Conj_mul_SNo_distrR__33__1)

SNo (x * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 0b3a47... and proposition id is 37aa39...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__33__10

L2991

Variable x : set

(*** Conj_mul_SNo_distrR__33__10 TMUb2jKZW6uy4A84fu72PmAsbpEco692EgM bounty of about 25 bars ***)

L2992

Variable y : set

L2993

Variable z : set

L2994

Variable w : set

L2995

Variable u : set

L2996

Variable v : set

L2997

Hypothesis H0 : SNo x

L2998

Hypothesis H1 : SNo y

L2999

Hypothesis H2 : SNo z

L3000

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3001

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3002

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3003

Hypothesis H6 : SNo (x + y)

L3004

Hypothesis H7 : SNo ((x + y) * z)

L3005

Hypothesis H8 : SNo (y * z)

L3006

Hypothesis H9 : SNo w

L3007

Hypothesis H11 : v ∈ SNoR z

L3008

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L3009

Hypothesis H13 : SNo u

L3010

Hypothesis H14 : u < x

L3011

Hypothesis H15 : SNo v

L3012

Hypothesis H16 : z < v

L3013

Hypothesis H17 : SNo (u * v)

L3014

Hypothesis H18 : SNo (u + y)

L3015

Hypothesis H19 : SNo (w + u * v)

L3016

Hypothesis H20 : SNo ((x + y) * v)

L3017

Hypothesis H21 : SNo (u * z)

L3018

Theorem. (Conj_mul_SNo_distrR__33__10)

SNo (x * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 8d2a42... and proposition id is 3fdaae...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__33__11

L3024

Variable x : set

(*** Conj_mul_SNo_distrR__33__11 TMWM3tmXyTq2VqGo2nuvKKqwaViUMeFpfcH bounty of about 25 bars ***)

L3025

Variable y : set

L3026

Variable z : set

L3027

Variable w : set

L3028

Variable u : set

L3029

Variable v : set

L3030

Hypothesis H0 : SNo x

L3031

Hypothesis H1 : SNo y

L3032

Hypothesis H2 : SNo z

L3033

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3034

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3035

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3036

Hypothesis H6 : SNo (x + y)

L3037

Hypothesis H7 : SNo ((x + y) * z)

L3038

Hypothesis H8 : SNo (y * z)

L3039

Hypothesis H9 : SNo w

L3040

Hypothesis H10 : u ∈ SNoL x

L3041

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L3042

Hypothesis H13 : SNo u

L3043

Hypothesis H14 : u < x

L3044

Hypothesis H15 : SNo v

L3045

Hypothesis H16 : z < v

L3046

Hypothesis H17 : SNo (u * v)

L3047

Hypothesis H18 : SNo (u + y)

L3048

Hypothesis H19 : SNo (w + u * v)

L3049

Hypothesis H20 : SNo ((x + y) * v)

L3050

Hypothesis H21 : SNo (u * z)

L3051

Theorem. (Conj_mul_SNo_distrR__33__11)

SNo (x * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is c86354... and proposition id is adcb15...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__34__15

L3057

Variable x : set

(*** Conj_mul_SNo_distrR__34__15 TMbuqEmUYukwC6DNGFGx2BiyFoXmdneaFda bounty of about 25 bars ***)

L3058

Variable y : set

L3059

Variable z : set

L3060

Variable w : set

L3061

Variable u : set

L3062

Variable v : set

L3063

Hypothesis H0 : SNo x

L3064

Hypothesis H1 : SNo y

L3065

Hypothesis H2 : SNo z

L3066

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3067

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3068

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3069

Hypothesis H6 : SNo (x + y)

L3070

Hypothesis H7 : SNo ((x + y) * z)

L3071

Hypothesis H8 : SNo (y * z)

L3072

Hypothesis H9 : SNo w

L3073

Hypothesis H10 : u ∈ SNoL x

L3074

Hypothesis H11 : v ∈ SNoR z

L3075

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L3076

Hypothesis H13 : SNo u

L3077

Hypothesis H14 : u < x

L3078

Hypothesis H16 : z < v

L3079

Hypothesis H17 : SNo (u * v)

L3080

Hypothesis H18 : SNo (u + y)

L3081

Hypothesis H19 : SNo (w + u * v)

L3082

Hypothesis H20 : SNo ((x + y) * v)

L3083

Theorem. (Conj_mul_SNo_distrR__34__15)

SNo (u * z) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is fab689... and proposition id is c652a9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__35__17

L3089

Variable x : set

(*** Conj_mul_SNo_distrR__35__17 TMRYkZvQTioECxVCcJCbkbeF2GpbtG2pQ5g bounty of about 25 bars ***)

L3090

Variable y : set

L3091

Variable z : set

L3092

Variable w : set

L3093

Variable u : set

L3094

Variable v : set

L3095

Hypothesis H0 : SNo x

L3096

Hypothesis H1 : SNo y

L3097

Hypothesis H2 : SNo z

L3098

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3099

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3100

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3101

Hypothesis H6 : SNo (x + y)

L3102

Hypothesis H7 : SNo ((x + y) * z)

L3103

Hypothesis H8 : SNo (y * z)

L3104

Hypothesis H9 : SNo w

L3105

Hypothesis H10 : u ∈ SNoL x

L3106

Hypothesis H11 : v ∈ SNoR z

L3107

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L3108

Hypothesis H13 : SNo u

L3109

Hypothesis H14 : u < x

L3110

Hypothesis H15 : SNo v

L3111

Hypothesis H16 : z < v

L3112

Hypothesis H18 : SNo (u + y)

L3113

Hypothesis H19 : SNo (w + u * v)

L3114

Theorem. (Conj_mul_SNo_distrR__35__17)

SNo ((x + y) * v) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 86b499... and proposition id is 812b3c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__36__16

L3120

Variable x : set

(*** Conj_mul_SNo_distrR__36__16 TMdYXjeNRGtQVG4koBEXz4mu1iWomVAQ1jC bounty of about 25 bars ***)

L3121

Variable y : set

L3122

Variable z : set

L3123

Variable w : set

L3124

Variable u : set

L3125

Variable v : set

L3126

Hypothesis H0 : SNo x

L3127

Hypothesis H1 : SNo y

L3128

Hypothesis H2 : SNo z

L3129

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3130

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3131

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3132

Hypothesis H6 : SNo (x + y)

L3133

Hypothesis H7 : SNo ((x + y) * z)

L3134

Hypothesis H8 : SNo (y * z)

L3135

Hypothesis H9 : SNo w

L3136

Hypothesis H10 : u ∈ SNoL x

L3137

Hypothesis H11 : v ∈ SNoR z

L3138

Hypothesis H12 : (u * z + x * v) ≤ w + u * v

L3139

Hypothesis H13 : SNo u

L3140

Hypothesis H14 : u < x

L3141

Hypothesis H15 : SNo v

L3142

Hypothesis H17 : SNo (u * v)

L3143

Hypothesis H18 : SNo (u + y)

L3144

Theorem. (Conj_mul_SNo_distrR__36__16)

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

In Proofgold the corresponding term root is f55de4... and proposition id is f21ef4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__37__12

L3150

Variable x : set

(*** Conj_mul_SNo_distrR__37__12 TMdvk5o3ry4bBYeevH9yHHVot9tnzYbgVn2 bounty of about 25 bars ***)

L3151

Variable y : set

L3152

Variable z : set

L3153

Variable w : set

L3154

Variable u : set

L3155

Variable v : set

L3156

Hypothesis H0 : SNo x

L3157

Hypothesis H1 : SNo y

L3158

Hypothesis H2 : SNo z

L3159

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3160

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3161

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3162

Hypothesis H6 : SNo (x + y)

L3163

Hypothesis H7 : SNo ((x + y) * z)

L3164

Hypothesis H8 : SNo (y * z)

L3165

Hypothesis H9 : SNo w

L3166

Hypothesis H10 : u ∈ SNoL x

L3167

Hypothesis H11 : v ∈ SNoR z

L3168

Hypothesis H13 : SNo u

L3169

Hypothesis H14 : u < x

L3170

Hypothesis H15 : SNo v

L3171

Hypothesis H16 : z < v

L3172

Hypothesis H17 : SNo (u * v)

L3173

Theorem. (Conj_mul_SNo_distrR__37__12)

SNo (u + y) → (x + y) * z < w + y * z

In Proofgold the corresponding term root is 0fa016... and proposition id is 94daac...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__39__5

L3179

Variable x : set

(*** Conj_mul_SNo_distrR__39__5 TMR3nD8zMeFFXiKFfhBkF6jm1hsSYepHH8y bounty of about 25 bars ***)

L3180

Variable y : set

L3181

Variable z : set

L3182

Variable w : set

L3183

Variable u : set

L3184

Variable v : set

L3185

Hypothesis H0 : SNo x

L3186

Hypothesis H1 : SNo y

L3187

Hypothesis H2 : SNo z

L3188

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3189

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3190

Hypothesis H6 : SNo (x + y)

L3191

Hypothesis H7 : SNo ((x + y) * z)

L3192

Hypothesis H8 : SNo (x * z)

L3193

Hypothesis H9 : SNo w

L3194

Hypothesis H10 : u ∈ SNoR y

L3195

Hypothesis H11 : v ∈ SNoR z

L3196

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3197

Hypothesis H13 : SNo u

L3198

Hypothesis H14 : y < u

L3199

Hypothesis H15 : SNo v

L3200

Hypothesis H16 : z < v

L3201

Hypothesis H17 : SNo (u * v)

L3202

Hypothesis H18 : SNo (x + u)

L3203

Hypothesis H19 : SNo (w + u * v)

L3204

Hypothesis H20 : SNo ((x + y) * v)

L3205

Hypothesis H21 : SNo (u * z)

L3206

Hypothesis H22 : SNo (x * v)

L3207

Hypothesis H23 : SNo (y * v)

L3208

Hypothesis H24 : SNo (w + x * z)

L3209

Hypothesis H25 : SNo (u * v + x * v)

L3210

Theorem. (Conj_mul_SNo_distrR__39__5)

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

In Proofgold the corresponding term root is 160480... and proposition id is e043c0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__40__1

L3216

Variable x : set

(*** Conj_mul_SNo_distrR__40__1 TMQUCDYK85o3twi18wq9L4DgUKi6gZtcPFC bounty of about 25 bars ***)

L3217

Variable y : set

L3218

Variable z : set

L3219

Variable w : set

L3220

Variable u : set

L3221

Variable v : set

L3222

Hypothesis H0 : SNo x

L3223

Hypothesis H2 : SNo z

L3224

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3225

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3226

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3227

Hypothesis H6 : SNo (x + y)

L3228

Hypothesis H7 : SNo ((x + y) * z)

L3229

Hypothesis H8 : SNo (x * z)

L3230

Hypothesis H9 : SNo w

L3231

Hypothesis H10 : u ∈ SNoR y

L3232

Hypothesis H11 : v ∈ SNoR z

L3233

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3234

Hypothesis H13 : SNo u

L3235

Hypothesis H14 : y < u

L3236

Hypothesis H15 : SNo v

L3237

Hypothesis H16 : z < v

L3238

Hypothesis H17 : SNo (u * v)

L3239

Hypothesis H18 : SNo (x + u)

L3240

Hypothesis H19 : SNo (w + u * v)

L3241

Hypothesis H20 : SNo ((x + y) * v)

L3242

Hypothesis H21 : SNo (u * z)

L3243

Hypothesis H22 : SNo (x * v)

L3244

Hypothesis H23 : SNo (y * v)

L3245

Hypothesis H24 : SNo (w + x * z)

L3246

Theorem. (Conj_mul_SNo_distrR__40__1)

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

In Proofgold the corresponding term root is ab5b2f... and proposition id is 02e2a0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__40__24

L3252

Variable x : set

(*** Conj_mul_SNo_distrR__40__24 TMS113RJw2iS1i5P74KVdgMNtYan1Ctuefb bounty of about 25 bars ***)

L3253

Variable y : set

L3254

Variable z : set

L3255

Variable w : set

L3256

Variable u : set

L3257

Variable v : set

L3258

Hypothesis H0 : SNo x

L3259

Hypothesis H1 : SNo y

L3260

Hypothesis H2 : SNo z

L3261

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3262

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3263

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3264

Hypothesis H6 : SNo (x + y)

L3265

Hypothesis H7 : SNo ((x + y) * z)

L3266

Hypothesis H8 : SNo (x * z)

L3267

Hypothesis H9 : SNo w

L3268

Hypothesis H10 : u ∈ SNoR y

L3269

Hypothesis H11 : v ∈ SNoR z

L3270

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3271

Hypothesis H13 : SNo u

L3272

Hypothesis H14 : y < u

L3273

Hypothesis H15 : SNo v

L3274

Hypothesis H16 : z < v

L3275

Hypothesis H17 : SNo (u * v)

L3276

Hypothesis H18 : SNo (x + u)

L3277

Hypothesis H19 : SNo (w + u * v)

L3278

Hypothesis H20 : SNo ((x + y) * v)

L3279

Hypothesis H21 : SNo (u * z)

L3280

Hypothesis H22 : SNo (x * v)

L3281

Hypothesis H23 : SNo (y * v)

L3282

Theorem. (Conj_mul_SNo_distrR__40__24)

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

In Proofgold the corresponding term root is 68d3f3... and proposition id is 9f3781...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__41__10

L3288

Variable x : set

(*** Conj_mul_SNo_distrR__41__10 TMZSG48dCkEuxXB7kbpaTTkfrJ6v9Tzwwf6 bounty of about 25 bars ***)

L3289

Variable y : set

L3290

Variable z : set

L3291

Variable w : set

L3292

Variable u : set

L3293

Variable v : set

L3294

Hypothesis H0 : SNo x

L3295

Hypothesis H1 : SNo y

L3296

Hypothesis H2 : SNo z

L3297

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3298

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3299

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3300

Hypothesis H6 : SNo (x + y)

L3301

Hypothesis H7 : SNo ((x + y) * z)

L3302

Hypothesis H8 : SNo (x * z)

L3303

Hypothesis H9 : SNo w

L3304

Hypothesis H11 : v ∈ SNoR z

L3305

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3306

Hypothesis H13 : SNo u

L3307

Hypothesis H14 : y < u

L3308

Hypothesis H15 : SNo v

L3309

Hypothesis H16 : z < v

L3310

Hypothesis H17 : SNo (u * v)

L3311

Hypothesis H18 : SNo (x + u)

L3312

Hypothesis H19 : SNo (w + u * v)

L3313

Hypothesis H20 : SNo ((x + y) * v)

L3314

Hypothesis H21 : SNo (u * z)

L3315

Hypothesis H22 : SNo (x * v)

L3316

Hypothesis H23 : SNo (y * v)

L3317

Theorem. (Conj_mul_SNo_distrR__41__10)

SNo (w + x * z) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 09027f... and proposition id is 9768e3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__41__14

L3323

Variable x : set

(*** Conj_mul_SNo_distrR__41__14 TMP6fgSW1qTQHd72h7CoGo9ofvcUCXpgxT8 bounty of about 25 bars ***)

L3324

Variable y : set

L3325

Variable z : set

L3326

Variable w : set

L3327

Variable u : set

L3328

Variable v : set

L3329

Hypothesis H0 : SNo x

L3330

Hypothesis H1 : SNo y

L3331

Hypothesis H2 : SNo z

L3332

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3333

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3334

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3335

Hypothesis H6 : SNo (x + y)

L3336

Hypothesis H7 : SNo ((x + y) * z)

L3337

Hypothesis H8 : SNo (x * z)

L3338

Hypothesis H9 : SNo w

L3339

Hypothesis H10 : u ∈ SNoR y

L3340

Hypothesis H11 : v ∈ SNoR z

L3341

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3342

Hypothesis H13 : SNo u

L3343

Hypothesis H15 : SNo v

L3344

Hypothesis H16 : z < v

L3345

Hypothesis H17 : SNo (u * v)

L3346

Hypothesis H18 : SNo (x + u)

L3347

Hypothesis H19 : SNo (w + u * v)

L3348

Hypothesis H20 : SNo ((x + y) * v)

L3349

Hypothesis H21 : SNo (u * z)

L3350

Hypothesis H22 : SNo (x * v)

L3351

Hypothesis H23 : SNo (y * v)

L3352

Theorem. (Conj_mul_SNo_distrR__41__14)

SNo (w + x * z) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is b3cc82... and proposition id is d483e6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__42__19

L3358

Variable x : set

(*** Conj_mul_SNo_distrR__42__19 TMLKqQ65FXqBmdoWQZ9DgixHHUSAswkCeF4 bounty of about 25 bars ***)

L3359

Variable y : set

L3360

Variable z : set

L3361

Variable w : set

L3362

Variable u : set

L3363

Variable v : set

L3364

Hypothesis H0 : SNo x

L3365

Hypothesis H1 : SNo y

L3366

Hypothesis H2 : SNo z

L3367

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3368

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3369

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3370

Hypothesis H6 : SNo (x + y)

L3371

Hypothesis H7 : SNo ((x + y) * z)

L3372

Hypothesis H8 : SNo (x * z)

L3373

Hypothesis H9 : SNo w

L3374

Hypothesis H10 : u ∈ SNoR y

L3375

Hypothesis H11 : v ∈ SNoR z

L3376

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3377

Hypothesis H13 : SNo u

L3378

Hypothesis H14 : y < u

L3379

Hypothesis H15 : SNo v

L3380

Hypothesis H16 : z < v

L3381

Hypothesis H17 : SNo (u * v)

L3382

Hypothesis H18 : SNo (x + u)

L3383

Hypothesis H20 : SNo ((x + y) * v)

L3384

Hypothesis H21 : SNo (u * z)

L3385

Hypothesis H22 : SNo (x * v)

L3386

Theorem. (Conj_mul_SNo_distrR__42__19)

SNo (y * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is b355ab... and proposition id is e518cb...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__44__4

L3392

Variable x : set

(*** Conj_mul_SNo_distrR__44__4 TMWGfq1LbAeFx4bHjLqm8ZoG7pBgJ5V3C3U bounty of about 25 bars ***)

L3393

Variable y : set

L3394

Variable z : set

L3395

Variable w : set

L3396

Variable u : set

L3397

Variable v : set

L3398

Hypothesis H0 : SNo x

L3399

Hypothesis H1 : SNo y

L3400

Hypothesis H2 : SNo z

L3401

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3402

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3403

Hypothesis H6 : SNo (x + y)

L3404

Hypothesis H7 : SNo ((x + y) * z)

L3405

Hypothesis H8 : SNo (x * z)

L3406

Hypothesis H9 : SNo w

L3407

Hypothesis H10 : u ∈ SNoR y

L3408

Hypothesis H11 : v ∈ SNoR z

L3409

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3410

Hypothesis H13 : SNo u

L3411

Hypothesis H14 : y < u

L3412

Hypothesis H15 : SNo v

L3413

Hypothesis H16 : z < v

L3414

Hypothesis H17 : SNo (u * v)

L3415

Hypothesis H18 : SNo (x + u)

L3416

Hypothesis H19 : SNo (w + u * v)

L3417

Hypothesis H20 : SNo ((x + y) * v)

L3418

Theorem. (Conj_mul_SNo_distrR__44__4)

SNo (u * z) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is dacdd1... and proposition id is 0d36e4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__45__16

L3424

Variable x : set

(*** Conj_mul_SNo_distrR__45__16 TMHLPJx6ynfxXTkzoWH5ofFdiHmgsPVqsbW bounty of about 25 bars ***)

L3425

Variable y : set

L3426

Variable z : set

L3427

Variable w : set

L3428

Variable u : set

L3429

Variable v : set

L3430

Hypothesis H0 : SNo x

L3431

Hypothesis H1 : SNo y

L3432

Hypothesis H2 : SNo z

L3433

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3434

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3435

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3436

Hypothesis H6 : SNo (x + y)

L3437

Hypothesis H7 : SNo ((x + y) * z)

L3438

Hypothesis H8 : SNo (x * z)

L3439

Hypothesis H9 : SNo w

L3440

Hypothesis H10 : u ∈ SNoR y

L3441

Hypothesis H11 : v ∈ SNoR z

L3442

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3443

Hypothesis H13 : SNo u

L3444

Hypothesis H14 : y < u

L3445

Hypothesis H15 : SNo v

L3446

Hypothesis H17 : SNo (u * v)

L3447

Hypothesis H18 : SNo (x + u)

L3448

Hypothesis H19 : SNo (w + u * v)

L3449

Theorem. (Conj_mul_SNo_distrR__45__16)

SNo ((x + y) * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 3d3ce7... and proposition id is 1e50b1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__47__17

L3455

Variable x : set

(*** Conj_mul_SNo_distrR__47__17 TMUyuaVyLYpWZ59z93sbXEMjs98kpWpanp7 bounty of about 25 bars ***)

L3456

Variable y : set

L3457

Variable z : set

L3458

Variable w : set

L3459

Variable u : set

L3460

Variable v : set

L3461

Hypothesis H0 : SNo x

L3462

Hypothesis H1 : SNo y

L3463

Hypothesis H2 : SNo z

L3464

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3465

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3466

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3467

Hypothesis H6 : SNo (x + y)

L3468

Hypothesis H7 : SNo ((x + y) * z)

L3469

Hypothesis H8 : SNo (x * z)

L3470

Hypothesis H9 : SNo w

L3471

Hypothesis H10 : u ∈ SNoR y

L3472

Hypothesis H11 : v ∈ SNoR z

L3473

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3474

Hypothesis H13 : SNo u

L3475

Hypothesis H14 : y < u

L3476

Hypothesis H15 : SNo v

L3477

Hypothesis H16 : z < v

L3478

Theorem. (Conj_mul_SNo_distrR__47__17)

SNo (x + u) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 8e9069... and proposition id is 43da1d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__48__11

L3484

Variable x : set

(*** Conj_mul_SNo_distrR__48__11 TMaz44LnE1pb9RaEGi5AGLx4g1WG1f1bptq bounty of about 25 bars ***)

L3485

Variable y : set

L3486

Variable z : set

L3487

Variable w : set

L3488

Variable u : set

L3489

Variable v : set

L3490

Hypothesis H0 : SNo x

L3491

Hypothesis H1 : SNo y

L3492

Hypothesis H2 : SNo z

L3493

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3494

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3495

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3496

Hypothesis H6 : SNo (x + y)

L3497

Hypothesis H7 : SNo ((x + y) * z)

L3498

Hypothesis H8 : SNo (x * z)

L3499

Hypothesis H9 : SNo w

L3500

Hypothesis H10 : u ∈ SNoR y

L3501

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3502

Hypothesis H13 : SNo u

L3503

Hypothesis H14 : y < u

L3504

Hypothesis H15 : SNo v

L3505

Hypothesis H16 : z < v

L3506

Theorem. (Conj_mul_SNo_distrR__48__11)

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

In Proofgold the corresponding term root is c11228... and proposition id is c27a91...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__48__15

L3512

Variable x : set

(*** Conj_mul_SNo_distrR__48__15 TMU4ADx8d4eAtmHruNTko9Anu4MrwNHFJpJ bounty of about 25 bars ***)

L3513

Variable y : set

L3514

Variable z : set

L3515

Variable w : set

L3516

Variable u : set

L3517

Variable v : set

L3518

Hypothesis H0 : SNo x

L3519

Hypothesis H1 : SNo y

L3520

Hypothesis H2 : SNo z

L3521

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3522

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3523

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3524

Hypothesis H6 : SNo (x + y)

L3525

Hypothesis H7 : SNo ((x + y) * z)

L3526

Hypothesis H8 : SNo (x * z)

L3527

Hypothesis H9 : SNo w

L3528

Hypothesis H10 : u ∈ SNoR y

L3529

Hypothesis H11 : v ∈ SNoR z

L3530

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3531

Hypothesis H13 : SNo u

L3532

Hypothesis H14 : y < u

L3533

Hypothesis H16 : z < v

L3534

Theorem. (Conj_mul_SNo_distrR__48__15)

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

In Proofgold the corresponding term root is 3c1a81... and proposition id is 5da8ca...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__49__2

L3540

Variable x : set

(*** Conj_mul_SNo_distrR__49__2 TMTZVPBF5BsrRLPKKBJc3dLpAvUitc2kwpP bounty of about 25 bars ***)

L3541

Variable y : set

L3542

Variable z : set

L3543

Variable w : set

L3544

Variable u : set

L3545

Variable v : set

L3546

Hypothesis H0 : SNo x

L3547

Hypothesis H1 : SNo y

L3548

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3549

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3550

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3551

Hypothesis H6 : SNo (x + y)

L3552

Hypothesis H7 : SNo ((x + y) * z)

L3553

Hypothesis H8 : SNo (x * z)

L3554

Hypothesis H9 : SNo w

L3555

Hypothesis H10 : u ∈ SNoL y

L3556

Hypothesis H11 : v ∈ SNoL z

L3557

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3558

Hypothesis H13 : SNo u

L3559

Hypothesis H14 : u < y

L3560

Hypothesis H15 : SNo v

L3561

Hypothesis H16 : v < z

L3562

Hypothesis H17 : SNo (u * v)

L3563

Hypothesis H18 : SNo (x + u)

L3564

Hypothesis H19 : SNo (w + u * v)

L3565

Hypothesis H20 : SNo ((x + y) * v)

L3566

Hypothesis H21 : SNo (u * z)

L3567

Hypothesis H22 : SNo (x * v)

L3568

Hypothesis H23 : SNo (y * v)

L3569

Hypothesis H24 : SNo (w + x * z)

L3570

Theorem. (Conj_mul_SNo_distrR__49__2)

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

In Proofgold the corresponding term root is 44282d... and proposition id is 0a4ef2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__52__2

L3576

Variable x : set

(*** Conj_mul_SNo_distrR__52__2 TMKdWbsrJMU2vPPzCWXTApXppKARV7Gt41R bounty of about 25 bars ***)

L3577

Variable y : set

L3578

Variable z : set

L3579

Variable w : set

L3580

Variable u : set

L3581

Variable v : set

L3582

Hypothesis H0 : SNo x

L3583

Hypothesis H1 : SNo y

L3584

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3585

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3586

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3587

Hypothesis H6 : SNo (x + y)

L3588

Hypothesis H7 : SNo ((x + y) * z)

L3589

Hypothesis H8 : SNo (x * z)

L3590

Hypothesis H9 : SNo w

L3591

Hypothesis H10 : u ∈ SNoL y

L3592

Hypothesis H11 : v ∈ SNoL z

L3593

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3594

Hypothesis H13 : SNo u

L3595

Hypothesis H14 : u < y

L3596

Hypothesis H15 : SNo v

L3597

Hypothesis H16 : v < z

L3598

Hypothesis H17 : SNo (u * v)

L3599

Hypothesis H18 : SNo (x + u)

L3600

Hypothesis H19 : SNo (w + u * v)

L3601

Hypothesis H20 : SNo ((x + y) * v)

L3602

Hypothesis H21 : SNo (u * z)

L3603

Theorem. (Conj_mul_SNo_distrR__52__2)

SNo (x * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 810d85... and proposition id is b16a81...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__52__9

L3609

Variable x : set

(*** Conj_mul_SNo_distrR__52__9 TMPnsZYKio5eknaCWc5sX8S74DsM9etm4gw bounty of about 25 bars ***)

L3610

Variable y : set

L3611

Variable z : set

L3612

Variable w : set

L3613

Variable u : set

L3614

Variable v : set

L3615

Hypothesis H0 : SNo x

L3616

Hypothesis H1 : SNo y

L3617

Hypothesis H2 : SNo z

L3618

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3619

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3620

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3621

Hypothesis H6 : SNo (x + y)

L3622

Hypothesis H7 : SNo ((x + y) * z)

L3623

Hypothesis H8 : SNo (x * z)

L3624

Hypothesis H10 : u ∈ SNoL y

L3625

Hypothesis H11 : v ∈ SNoL z

L3626

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3627

Hypothesis H13 : SNo u

L3628

Hypothesis H14 : u < y

L3629

Hypothesis H15 : SNo v

L3630

Hypothesis H16 : v < z

L3631

Hypothesis H17 : SNo (u * v)

L3632

Hypothesis H18 : SNo (x + u)

L3633

Hypothesis H19 : SNo (w + u * v)

L3634

Hypothesis H20 : SNo ((x + y) * v)

L3635

Hypothesis H21 : SNo (u * z)

L3636

Theorem. (Conj_mul_SNo_distrR__52__9)

SNo (x * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 67dc2f... and proposition id is 9395a3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__52__13

L3642

Variable x : set

(*** Conj_mul_SNo_distrR__52__13 TMWAuogoqYeNYe8ePKXTF4LnMBFCnR1FLyY bounty of about 25 bars ***)

L3643

Variable y : set

L3644

Variable z : set

L3645

Variable w : set

L3646

Variable u : set

L3647

Variable v : set

L3648

Hypothesis H0 : SNo x

L3649

Hypothesis H1 : SNo y

L3650

Hypothesis H2 : SNo z

L3651

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3652

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3653

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3654

Hypothesis H6 : SNo (x + y)

L3655

Hypothesis H7 : SNo ((x + y) * z)

L3656

Hypothesis H8 : SNo (x * z)

L3657

Hypothesis H9 : SNo w

L3658

Hypothesis H10 : u ∈ SNoL y

L3659

Hypothesis H11 : v ∈ SNoL z

L3660

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3661

Hypothesis H14 : u < y

L3662

Hypothesis H15 : SNo v

L3663

Hypothesis H16 : v < z

L3664

Hypothesis H17 : SNo (u * v)

L3665

Hypothesis H18 : SNo (x + u)

L3666

Hypothesis H19 : SNo (w + u * v)

L3667

Hypothesis H20 : SNo ((x + y) * v)

L3668

Hypothesis H21 : SNo (u * z)

L3669

Theorem. (Conj_mul_SNo_distrR__52__13)

SNo (x * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 7b5791... and proposition id is 57bb75...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__52__19

L3675

Variable x : set

(*** Conj_mul_SNo_distrR__52__19 TMH7wgEZFjYczdFvAs18ukb8d9fewfFrnbu bounty of about 25 bars ***)

L3676

Variable y : set

L3677

Variable z : set

L3678

Variable w : set

L3679

Variable u : set

L3680

Variable v : set

L3681

Hypothesis H0 : SNo x

L3682

Hypothesis H1 : SNo y

L3683

Hypothesis H2 : SNo z

L3684

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3685

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3686

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3687

Hypothesis H6 : SNo (x + y)

L3688

Hypothesis H7 : SNo ((x + y) * z)

L3689

Hypothesis H8 : SNo (x * z)

L3690

Hypothesis H9 : SNo w

L3691

Hypothesis H10 : u ∈ SNoL y

L3692

Hypothesis H11 : v ∈ SNoL z

L3693

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3694

Hypothesis H13 : SNo u

L3695

Hypothesis H14 : u < y

L3696

Hypothesis H15 : SNo v

L3697

Hypothesis H16 : v < z

L3698

Hypothesis H17 : SNo (u * v)

L3699

Hypothesis H18 : SNo (x + u)

L3700

Hypothesis H20 : SNo ((x + y) * v)

L3701

Hypothesis H21 : SNo (u * z)

L3702

Theorem. (Conj_mul_SNo_distrR__52__19)

SNo (x * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 77c53a... and proposition id is 677e05...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__52__21

L3708

Variable x : set

(*** Conj_mul_SNo_distrR__52__21 TMVjuCjVmizyaEigPLu548NFq3eRivbhoyA bounty of about 25 bars ***)

L3709

Variable y : set

L3710

Variable z : set

L3711

Variable w : set

L3712

Variable u : set

L3713

Variable v : set

L3714

Hypothesis H0 : SNo x

L3715

Hypothesis H1 : SNo y

L3716

Hypothesis H2 : SNo z

L3717

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3718

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3719

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3720

Hypothesis H6 : SNo (x + y)

L3721

Hypothesis H7 : SNo ((x + y) * z)

L3722

Hypothesis H8 : SNo (x * z)

L3723

Hypothesis H9 : SNo w

L3724

Hypothesis H10 : u ∈ SNoL y

L3725

Hypothesis H11 : v ∈ SNoL z

L3726

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3727

Hypothesis H13 : SNo u

L3728

Hypothesis H14 : u < y

L3729

Hypothesis H15 : SNo v

L3730

Hypothesis H16 : v < z

L3731

Hypothesis H17 : SNo (u * v)

L3732

Hypothesis H18 : SNo (x + u)

L3733

Hypothesis H19 : SNo (w + u * v)

L3734

Hypothesis H20 : SNo ((x + y) * v)

L3735

Theorem. (Conj_mul_SNo_distrR__52__21)

SNo (x * v) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is b9e38e... and proposition id is 330362...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__53__19

L3741

Variable x : set

(*** Conj_mul_SNo_distrR__53__19 TMd77WxRGj57fcLRo7Y4VnN2wJ9VieG9fgp bounty of about 25 bars ***)

L3742

Variable y : set

L3743

Variable z : set

L3744

Variable w : set

L3745

Variable u : set

L3746

Variable v : set

L3747

Hypothesis H0 : SNo x

L3748

Hypothesis H1 : SNo y

L3749

Hypothesis H2 : SNo z

L3750

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3751

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3752

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3753

Hypothesis H6 : SNo (x + y)

L3754

Hypothesis H7 : SNo ((x + y) * z)

L3755

Hypothesis H8 : SNo (x * z)

L3756

Hypothesis H9 : SNo w

L3757

Hypothesis H10 : u ∈ SNoL y

L3758

Hypothesis H11 : v ∈ SNoL z

L3759

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3760

Hypothesis H13 : SNo u

L3761

Hypothesis H14 : u < y

L3762

Hypothesis H15 : SNo v

L3763

Hypothesis H16 : v < z

L3764

Hypothesis H17 : SNo (u * v)

L3765

Hypothesis H18 : SNo (x + u)

L3766

Hypothesis H20 : SNo ((x + y) * v)

L3767

Theorem. (Conj_mul_SNo_distrR__53__19)

SNo (u * z) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 026d6d... and proposition id is 4bcb7f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__56__5

L3773

Variable x : set

(*** Conj_mul_SNo_distrR__56__5 TMWNj5NmkZjvpuQVcHCKriFEUNdBjUs21Pi bounty of about 25 bars ***)

L3774

Variable y : set

L3775

Variable z : set

L3776

Variable w : set

L3777

Variable u : set

L3778

Variable v : set

L3779

Hypothesis H0 : SNo x

L3780

Hypothesis H1 : SNo y

L3781

Hypothesis H2 : SNo z

L3782

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3783

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3784

Hypothesis H6 : SNo (x + y)

L3785

Hypothesis H7 : SNo ((x + y) * z)

L3786

Hypothesis H8 : SNo (x * z)

L3787

Hypothesis H9 : SNo w

L3788

Hypothesis H10 : u ∈ SNoL y

L3789

Hypothesis H11 : v ∈ SNoL z

L3790

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3791

Hypothesis H13 : SNo u

L3792

Hypothesis H14 : u < y

L3793

Hypothesis H15 : SNo v

L3794

Hypothesis H16 : v < z

L3795

Hypothesis H17 : SNo (u * v)

L3796

Theorem. (Conj_mul_SNo_distrR__56__5)

SNo (x + u) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 4da15c... and proposition id is cb475a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__56__15

L3802

Variable x : set

(*** Conj_mul_SNo_distrR__56__15 TMNftAcP2idEbztxAk7iNGPiwNKpKjBEgUA bounty of about 25 bars ***)

L3803

Variable y : set

L3804

Variable z : set

L3805

Variable w : set

L3806

Variable u : set

L3807

Variable v : set

L3808

Hypothesis H0 : SNo x

L3809

Hypothesis H1 : SNo y

L3810

Hypothesis H2 : SNo z

L3811

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3812

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3813

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3814

Hypothesis H6 : SNo (x + y)

L3815

Hypothesis H7 : SNo ((x + y) * z)

L3816

Hypothesis H8 : SNo (x * z)

L3817

Hypothesis H9 : SNo w

L3818

Hypothesis H10 : u ∈ SNoL y

L3819

Hypothesis H11 : v ∈ SNoL z

L3820

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3821

Hypothesis H13 : SNo u

L3822

Hypothesis H14 : u < y

L3823

Hypothesis H16 : v < z

L3824

Hypothesis H17 : SNo (u * v)

L3825

Theorem. (Conj_mul_SNo_distrR__56__15)

SNo (x + u) → (w + x * z) < (x + y) * z

In Proofgold the corresponding term root is 8c58a9... and proposition id is 29807c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__57__7

L3831

Variable x : set

(*** Conj_mul_SNo_distrR__57__7 TMWMo5S4S8uA1Xa94wAmho8QYLMQfGa6B11 bounty of about 25 bars ***)

L3832

Variable y : set

L3833

Variable z : set

L3834

Variable w : set

L3835

Variable u : set

L3836

Variable v : set

L3837

Hypothesis H0 : SNo x

L3838

Hypothesis H1 : SNo y

L3839

Hypothesis H2 : SNo z

L3840

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L3841

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3842

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L3843

Hypothesis H6 : SNo (x + y)

L3844

Hypothesis H8 : SNo (x * z)

L3845

Hypothesis H9 : SNo w

L3846

Hypothesis H10 : u ∈ SNoL y

L3847

Hypothesis H11 : v ∈ SNoL z

L3848

Hypothesis H12 : (w + u * v) ≤ u * z + y * v

L3849

Hypothesis H13 : SNo u

L3850

Hypothesis H14 : u < y

L3851

Hypothesis H15 : SNo v

L3852

Hypothesis H16 : v < z

L3853

Theorem. (Conj_mul_SNo_distrR__57__7)

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

In Proofgold the corresponding term root is 959fc6... and proposition id is ce5166...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__58__19

L3859

Variable x : set

(*** Conj_mul_SNo_distrR__58__19 TMGmFgPdFVZjvtefrPUoLPyy22zcyMf92Co bounty of about 25 bars ***)

L3860

Variable y : set

L3861

Variable z : set

L3862

Variable w : set

L3863

Variable u : set

L3864

Variable v : set

L3865

Hypothesis H0 : SNo x

L3866

Hypothesis H1 : SNo y

L3867

Hypothesis H2 : SNo z

L3868

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3869

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3870

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3871

Hypothesis H6 : SNo (x + y)

L3872

Hypothesis H7 : SNo ((x + y) * z)

L3873

Hypothesis H8 : SNo (y * z)

L3874

Hypothesis H9 : SNo w

L3875

Hypothesis H10 : u ∈ SNoR x

L3876

Hypothesis H11 : v ∈ SNoR z

L3877

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L3878

Hypothesis H13 : SNo u

L3879

Hypothesis H14 : x < u

L3880

Hypothesis H15 : SNo v

L3881

Hypothesis H16 : z < v

L3882

Hypothesis H17 : SNo (u * v)

L3883

Hypothesis H18 : SNo (u + y)

L3884

Hypothesis H20 : SNo ((x + y) * v)

L3885

Hypothesis H21 : SNo (u * z)

L3886

Hypothesis H22 : SNo (x * v)

L3887

Hypothesis H23 : SNo (y * v)

L3888

Hypothesis H24 : SNo (w + y * z)

L3889

Theorem. (Conj_mul_SNo_distrR__58__19)

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

In Proofgold the corresponding term root is 5f698c... and proposition id is 08f1f2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__60__18

L3895

Variable x : set

(*** Conj_mul_SNo_distrR__60__18 TMcF7xxLge6MoZsEJSwT7UVmBTX3GSaMVv9 bounty of about 25 bars ***)

L3896

Variable y : set

L3897

Variable z : set

L3898

Variable w : set

L3899

Variable u : set

L3900

Variable v : set

L3901

Hypothesis H0 : SNo x

L3902

Hypothesis H1 : SNo y

L3903

Hypothesis H2 : SNo z

L3904

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3905

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3906

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3907

Hypothesis H6 : SNo (x + y)

L3908

Hypothesis H7 : SNo ((x + y) * z)

L3909

Hypothesis H8 : SNo (y * z)

L3910

Hypothesis H9 : SNo w

L3911

Hypothesis H10 : u ∈ SNoR x

L3912

Hypothesis H11 : v ∈ SNoR z

L3913

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L3914

Hypothesis H13 : SNo u

L3915

Hypothesis H14 : x < u

L3916

Hypothesis H15 : SNo v

L3917

Hypothesis H16 : z < v

L3918

Hypothesis H17 : SNo (u * v)

L3919

Hypothesis H19 : SNo (w + u * v)

L3920

Hypothesis H20 : SNo ((x + y) * v)

L3921

Hypothesis H21 : SNo (u * z)

L3922

Hypothesis H22 : SNo (x * v)

L3923

Theorem. (Conj_mul_SNo_distrR__60__18)

SNo (y * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 432386... and proposition id is 7d99c0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__61__21

L3929

Variable x : set

(*** Conj_mul_SNo_distrR__61__21 TMdTGbgS4zY9qMjAkJnny7Kr8AtS28Se6UQ bounty of about 25 bars ***)

L3930

Variable y : set

L3931

Variable z : set

L3932

Variable w : set

L3933

Variable u : set

L3934

Variable v : set

L3935

Hypothesis H0 : SNo x

L3936

Hypothesis H1 : SNo y

L3937

Hypothesis H2 : SNo z

L3938

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3939

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3940

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3941

Hypothesis H6 : SNo (x + y)

L3942

Hypothesis H7 : SNo ((x + y) * z)

L3943

Hypothesis H8 : SNo (y * z)

L3944

Hypothesis H9 : SNo w

L3945

Hypothesis H10 : u ∈ SNoR x

L3946

Hypothesis H11 : v ∈ SNoR z

L3947

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L3948

Hypothesis H13 : SNo u

L3949

Hypothesis H14 : x < u

L3950

Hypothesis H15 : SNo v

L3951

Hypothesis H16 : z < v

L3952

Hypothesis H17 : SNo (u * v)

L3953

Hypothesis H18 : SNo (u + y)

L3954

Hypothesis H19 : SNo (w + u * v)

L3955

Hypothesis H20 : SNo ((x + y) * v)

L3956

Theorem. (Conj_mul_SNo_distrR__61__21)

SNo (x * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 51acc6... and proposition id is 0c86e3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__63__2

L3962

Variable x : set

(*** Conj_mul_SNo_distrR__63__2 TMW6MBPjUnTqFXsPGtfWJDU57Nc6qMuKgww bounty of about 25 bars ***)

L3963

Variable y : set

L3964

Variable z : set

L3965

Variable w : set

L3966

Variable u : set

L3967

Variable v : set

L3968

Hypothesis H0 : SNo x

L3969

Hypothesis H1 : SNo y

L3970

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L3971

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L3972

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L3973

Hypothesis H6 : SNo (x + y)

L3974

Hypothesis H7 : SNo ((x + y) * z)

L3975

Hypothesis H8 : SNo (y * z)

L3976

Hypothesis H9 : SNo w

L3977

Hypothesis H10 : u ∈ SNoR x

L3978

Hypothesis H11 : v ∈ SNoR z

L3979

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L3980

Hypothesis H13 : SNo u

L3981

Hypothesis H14 : x < u

L3982

Hypothesis H15 : SNo v

L3983

Hypothesis H16 : z < v

L3984

Hypothesis H17 : SNo (u * v)

L3985

Hypothesis H18 : SNo (u + y)

L3986

Hypothesis H19 : SNo (w + u * v)

L3987

Theorem. (Conj_mul_SNo_distrR__63__2)

SNo ((x + y) * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 2a5635... and proposition id is 2808fa...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__64__17

L3993

Variable x : set

(*** Conj_mul_SNo_distrR__64__17 TMRzoL3Y9GrmMz2XoGinfT43U4ZMX4Q1n9H bounty of about 25 bars ***)

L3994

Variable y : set

L3995

Variable z : set

L3996

Variable w : set

L3997

Variable u : set

L3998

Variable v : set

L3999

Hypothesis H0 : SNo x

L4000

Hypothesis H1 : SNo y

L4001

Hypothesis H2 : SNo z

L4002

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4003

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4004

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4005

Hypothesis H6 : SNo (x + y)

L4006

Hypothesis H7 : SNo ((x + y) * z)

L4007

Hypothesis H8 : SNo (y * z)

L4008

Hypothesis H9 : SNo w

L4009

Hypothesis H10 : u ∈ SNoR x

L4010

Hypothesis H11 : v ∈ SNoR z

L4011

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4012

Hypothesis H13 : SNo u

L4013

Hypothesis H14 : x < u

L4014

Hypothesis H15 : SNo v

L4015

Hypothesis H16 : z < v

L4016

Hypothesis H18 : SNo (u + y)

L4017

Theorem. (Conj_mul_SNo_distrR__64__17)

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

In Proofgold the corresponding term root is 663289... and proposition id is b668dd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__67__6

L4023

Variable x : set

(*** Conj_mul_SNo_distrR__67__6 TMKwPK1NSKTXSxa7P25qsYB1E4QvXkDxWdx bounty of about 25 bars ***)

L4024

Variable y : set

L4025

Variable z : set

L4026

Variable w : set

L4027

Variable u : set

L4028

Variable v : set

L4029

Hypothesis H0 : SNo x

L4030

Hypothesis H1 : SNo y

L4031

Hypothesis H2 : SNo z

L4032

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4033

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4034

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4035

Hypothesis H7 : SNo ((x + y) * z)

L4036

Hypothesis H8 : SNo (y * z)

L4037

Hypothesis H9 : SNo w

L4038

Hypothesis H10 : u ∈ SNoL x

L4039

Hypothesis H11 : v ∈ SNoL z

L4040

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4041

Hypothesis H13 : SNo u

L4042

Hypothesis H14 : u < x

L4043

Hypothesis H15 : SNo v

L4044

Hypothesis H16 : v < z

L4045

Hypothesis H17 : SNo (u * v)

L4046

Hypothesis H18 : SNo (u + y)

L4047

Hypothesis H19 : SNo (w + u * v)

L4048

Hypothesis H20 : SNo ((x + y) * v)

L4049

Hypothesis H21 : SNo (u * z)

L4050

Hypothesis H22 : SNo (x * v)

L4051

Hypothesis H23 : SNo (y * v)

L4052

Hypothesis H24 : SNo (w + y * z)

L4053

Hypothesis H25 : SNo (u * v + y * v)

L4054

Theorem. (Conj_mul_SNo_distrR__67__6)

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

In Proofgold the corresponding term root is 264f58... and proposition id is ab1c11...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__67__23

L4060

Variable x : set

(*** Conj_mul_SNo_distrR__67__23 TMUHGWg6EDJfHSmJCUtPgpQogEQWeZ4vYgJ bounty of about 25 bars ***)

L4061

Variable y : set

L4062

Variable z : set

L4063

Variable w : set

L4064

Variable u : set

L4065

Variable v : set

L4066

Hypothesis H0 : SNo x

L4067

Hypothesis H1 : SNo y

L4068

Hypothesis H2 : SNo z

L4069

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4070

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4071

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4072

Hypothesis H6 : SNo (x + y)

L4073

Hypothesis H7 : SNo ((x + y) * z)

L4074

Hypothesis H8 : SNo (y * z)

L4075

Hypothesis H9 : SNo w

L4076

Hypothesis H10 : u ∈ SNoL x

L4077

Hypothesis H11 : v ∈ SNoL z

L4078

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4079

Hypothesis H13 : SNo u

L4080

Hypothesis H14 : u < x

L4081

Hypothesis H15 : SNo v

L4082

Hypothesis H16 : v < z

L4083

Hypothesis H17 : SNo (u * v)

L4084

Hypothesis H18 : SNo (u + y)

L4085

Hypothesis H19 : SNo (w + u * v)

L4086

Hypothesis H20 : SNo ((x + y) * v)

L4087

Hypothesis H21 : SNo (u * z)

L4088

Hypothesis H22 : SNo (x * v)

L4089

Hypothesis H24 : SNo (w + y * z)

L4090

Hypothesis H25 : SNo (u * v + y * v)

L4091

Theorem. (Conj_mul_SNo_distrR__67__23)

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

In Proofgold the corresponding term root is c14e86... and proposition id is 2488f0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__68__11

L4097

Variable x : set

(*** Conj_mul_SNo_distrR__68__11 TMdn7oyQpfHuXPmUTHiRmP4yinV5WrvXEEy bounty of about 25 bars ***)

L4098

Variable y : set

L4099

Variable z : set

L4100

Variable w : set

L4101

Variable u : set

L4102

Variable v : set

L4103

Hypothesis H0 : SNo x

L4104

Hypothesis H1 : SNo y

L4105

Hypothesis H2 : SNo z

L4106

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4107

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4108

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4109

Hypothesis H6 : SNo (x + y)

L4110

Hypothesis H7 : SNo ((x + y) * z)

L4111

Hypothesis H8 : SNo (y * z)

L4112

Hypothesis H9 : SNo w

L4113

Hypothesis H10 : u ∈ SNoL x

L4114

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4115

Hypothesis H13 : SNo u

L4116

Hypothesis H14 : u < x

L4117

Hypothesis H15 : SNo v

L4118

Hypothesis H16 : v < z

L4119

Hypothesis H17 : SNo (u * v)

L4120

Hypothesis H18 : SNo (u + y)

L4121

Hypothesis H19 : SNo (w + u * v)

L4122

Hypothesis H20 : SNo ((x + y) * v)

L4123

Hypothesis H21 : SNo (u * z)

L4124

Hypothesis H22 : SNo (x * v)

L4125

Hypothesis H23 : SNo (y * v)

L4126

Hypothesis H24 : SNo (w + y * z)

L4127

Theorem. (Conj_mul_SNo_distrR__68__11)

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

In Proofgold the corresponding term root is 05b155... and proposition id is f25e21...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__68__13

L4133

Variable x : set

(*** Conj_mul_SNo_distrR__68__13 TMJtgS4CUtwWszuK7bu6TJrMqQAQRyJB8J7 bounty of about 25 bars ***)

L4134

Variable y : set

L4135

Variable z : set

L4136

Variable w : set

L4137

Variable u : set

L4138

Variable v : set

L4139

Hypothesis H0 : SNo x

L4140

Hypothesis H1 : SNo y

L4141

Hypothesis H2 : SNo z

L4142

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4143

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4144

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4145

Hypothesis H6 : SNo (x + y)

L4146

Hypothesis H7 : SNo ((x + y) * z)

L4147

Hypothesis H8 : SNo (y * z)

L4148

Hypothesis H9 : SNo w

L4149

Hypothesis H10 : u ∈ SNoL x

L4150

Hypothesis H11 : v ∈ SNoL z

L4151

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4152

Hypothesis H14 : u < x

L4153

Hypothesis H15 : SNo v

L4154

Hypothesis H16 : v < z

L4155

Hypothesis H17 : SNo (u * v)

L4156

Hypothesis H18 : SNo (u + y)

L4157

Hypothesis H19 : SNo (w + u * v)

L4158

Hypothesis H20 : SNo ((x + y) * v)

L4159

Hypothesis H21 : SNo (u * z)

L4160

Hypothesis H22 : SNo (x * v)

L4161

Hypothesis H23 : SNo (y * v)

L4162

Hypothesis H24 : SNo (w + y * z)

L4163

Theorem. (Conj_mul_SNo_distrR__68__13)

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

In Proofgold the corresponding term root is d5c959... and proposition id is 0e9c67...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__69__4

L4169

Variable x : set

(*** Conj_mul_SNo_distrR__69__4 TMbPgDJvxqmUnqd3yJGY5Fz6Q9SUUijkwE9 bounty of about 25 bars ***)

L4170

Variable y : set

L4171

Variable z : set

L4172

Variable w : set

L4173

Variable u : set

L4174

Variable v : set

L4175

Hypothesis H0 : SNo x

L4176

Hypothesis H1 : SNo y

L4177

Hypothesis H2 : SNo z

L4178

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4179

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4180

Hypothesis H6 : SNo (x + y)

L4181

Hypothesis H7 : SNo ((x + y) * z)

L4182

Hypothesis H8 : SNo (y * z)

L4183

Hypothesis H9 : SNo w

L4184

Hypothesis H10 : u ∈ SNoL x

L4185

Hypothesis H11 : v ∈ SNoL z

L4186

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4187

Hypothesis H13 : SNo u

L4188

Hypothesis H14 : u < x

L4189

Hypothesis H15 : SNo v

L4190

Hypothesis H16 : v < z

L4191

Hypothesis H17 : SNo (u * v)

L4192

Hypothesis H18 : SNo (u + y)

L4193

Hypothesis H19 : SNo (w + u * v)

L4194

Hypothesis H20 : SNo ((x + y) * v)

L4195

Hypothesis H21 : SNo (u * z)

L4196

Hypothesis H22 : SNo (x * v)

L4197

Hypothesis H23 : SNo (y * v)

L4198

Theorem. (Conj_mul_SNo_distrR__69__4)

SNo (w + y * z) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is c29acb... and proposition id is 567d28...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__69__8

L4204

Variable x : set

(*** Conj_mul_SNo_distrR__69__8 TMTis28vThZDvEsUqvr9uVLaW9X6o5a9o45 bounty of about 25 bars ***)

L4205

Variable y : set

L4206

Variable z : set

L4207

Variable w : set

L4208

Variable u : set

L4209

Variable v : set

L4210

Hypothesis H0 : SNo x

L4211

Hypothesis H1 : SNo y

L4212

Hypothesis H2 : SNo z

L4213

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4214

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4215

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4216

Hypothesis H6 : SNo (x + y)

L4217

Hypothesis H7 : SNo ((x + y) * z)

L4218

Hypothesis H9 : SNo w

L4219

Hypothesis H10 : u ∈ SNoL x

L4220

Hypothesis H11 : v ∈ SNoL z

L4221

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4222

Hypothesis H13 : SNo u

L4223

Hypothesis H14 : u < x

L4224

Hypothesis H15 : SNo v

L4225

Hypothesis H16 : v < z

L4226

Hypothesis H17 : SNo (u * v)

L4227

Hypothesis H18 : SNo (u + y)

L4228

Hypothesis H19 : SNo (w + u * v)

L4229

Hypothesis H20 : SNo ((x + y) * v)

L4230

Hypothesis H21 : SNo (u * z)

L4231

Hypothesis H22 : SNo (x * v)

L4232

Hypothesis H23 : SNo (y * v)

L4233

Theorem. (Conj_mul_SNo_distrR__69__8)

SNo (w + y * z) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 2f7f54... and proposition id is b86441...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__70__19

L4239

Variable x : set

(*** Conj_mul_SNo_distrR__70__19 TMR8eFinNYkj2hZx9erM3aK81RvujHiRvCu bounty of about 25 bars ***)

L4240

Variable y : set

L4241

Variable z : set

L4242

Variable w : set

L4243

Variable u : set

L4244

Variable v : set

L4245

Hypothesis H0 : SNo x

L4246

Hypothesis H1 : SNo y

L4247

Hypothesis H2 : SNo z

L4248

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4249

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4250

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4251

Hypothesis H6 : SNo (x + y)

L4252

Hypothesis H7 : SNo ((x + y) * z)

L4253

Hypothesis H8 : SNo (y * z)

L4254

Hypothesis H9 : SNo w

L4255

Hypothesis H10 : u ∈ SNoL x

L4256

Hypothesis H11 : v ∈ SNoL z

L4257

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4258

Hypothesis H13 : SNo u

L4259

Hypothesis H14 : u < x

L4260

Hypothesis H15 : SNo v

L4261

Hypothesis H16 : v < z

L4262

Hypothesis H17 : SNo (u * v)

L4263

Hypothesis H18 : SNo (u + y)

L4264

Hypothesis H20 : SNo ((x + y) * v)

L4265

Hypothesis H21 : SNo (u * z)

L4266

Hypothesis H22 : SNo (x * v)

L4267

Theorem. (Conj_mul_SNo_distrR__70__19)

SNo (y * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 93a3a5... and proposition id is 0608a4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__70__20

L4273

Variable x : set

(*** Conj_mul_SNo_distrR__70__20 TMJpqoHR8tBDSavGk9sD6458zGvBEjWadCa bounty of about 25 bars ***)

L4274

Variable y : set

L4275

Variable z : set

L4276

Variable w : set

L4277

Variable u : set

L4278

Variable v : set

L4279

Hypothesis H0 : SNo x

L4280

Hypothesis H1 : SNo y

L4281

Hypothesis H2 : SNo z

L4282

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4283

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4284

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4285

Hypothesis H6 : SNo (x + y)

L4286

Hypothesis H7 : SNo ((x + y) * z)

L4287

Hypothesis H8 : SNo (y * z)

L4288

Hypothesis H9 : SNo w

L4289

Hypothesis H10 : u ∈ SNoL x

L4290

Hypothesis H11 : v ∈ SNoL z

L4291

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4292

Hypothesis H13 : SNo u

L4293

Hypothesis H14 : u < x

L4294

Hypothesis H15 : SNo v

L4295

Hypothesis H16 : v < z

L4296

Hypothesis H17 : SNo (u * v)

L4297

Hypothesis H18 : SNo (u + y)

L4298

Hypothesis H19 : SNo (w + u * v)

L4299

Hypothesis H21 : SNo (u * z)

L4300

Hypothesis H22 : SNo (x * v)

L4301

Theorem. (Conj_mul_SNo_distrR__70__20)

SNo (y * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 7ad97b... and proposition id is 1a4651...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__71__13

L4307

Variable x : set

(*** Conj_mul_SNo_distrR__71__13 TMXnnCKVRhppcUsZGLjAjc6m3y19kqViVrG bounty of about 25 bars ***)

L4308

Variable y : set

L4309

Variable z : set

L4310

Variable w : set

L4311

Variable u : set

L4312

Variable v : set

L4313

Hypothesis H0 : SNo x

L4314

Hypothesis H1 : SNo y

L4315

Hypothesis H2 : SNo z

L4316

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4317

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4318

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4319

Hypothesis H6 : SNo (x + y)

L4320

Hypothesis H7 : SNo ((x + y) * z)

L4321

Hypothesis H8 : SNo (y * z)

L4322

Hypothesis H9 : SNo w

L4323

Hypothesis H10 : u ∈ SNoL x

L4324

Hypothesis H11 : v ∈ SNoL z

L4325

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4326

Hypothesis H14 : u < x

L4327

Hypothesis H15 : SNo v

L4328

Hypothesis H16 : v < z

L4329

Hypothesis H17 : SNo (u * v)

L4330

Hypothesis H18 : SNo (u + y)

L4331

Hypothesis H19 : SNo (w + u * v)

L4332

Hypothesis H20 : SNo ((x + y) * v)

L4333

Hypothesis H21 : SNo (u * z)

L4334

Theorem. (Conj_mul_SNo_distrR__71__13)

SNo (x * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is eebffe... and proposition id is f06a83...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__73__19

L4340

Variable x : set

(*** Conj_mul_SNo_distrR__73__19 TMRCkYVw8S41KhsMLC4Wq8bHrBLU3rrK7Ht bounty of about 25 bars ***)

L4341

Variable y : set

L4342

Variable z : set

L4343

Variable w : set

L4344

Variable u : set

L4345

Variable v : set

L4346

Hypothesis H0 : SNo x

L4347

Hypothesis H1 : SNo y

L4348

Hypothesis H2 : SNo z

L4349

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4350

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4351

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4352

Hypothesis H6 : SNo (x + y)

L4353

Hypothesis H7 : SNo ((x + y) * z)

L4354

Hypothesis H8 : SNo (y * z)

L4355

Hypothesis H9 : SNo w

L4356

Hypothesis H10 : u ∈ SNoL x

L4357

Hypothesis H11 : v ∈ SNoL z

L4358

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4359

Hypothesis H13 : SNo u

L4360

Hypothesis H14 : u < x

L4361

Hypothesis H15 : SNo v

L4362

Hypothesis H16 : v < z

L4363

Hypothesis H17 : SNo (u * v)

L4364

Hypothesis H18 : SNo (u + y)

L4365

Theorem. (Conj_mul_SNo_distrR__73__19)

SNo ((x + y) * v) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 64f42c... and proposition id is c5aff0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__74__5

L4371

Variable x : set

(*** Conj_mul_SNo_distrR__74__5 TMUPB9pzygmuQUd64T2mbg9We8K7Sx8S84r bounty of about 25 bars ***)

L4372

Variable y : set

L4373

Variable z : set

L4374

Variable w : set

L4375

Variable u : set

L4376

Variable v : set

L4377

Hypothesis H0 : SNo x

L4378

Hypothesis H1 : SNo y

L4379

Hypothesis H2 : SNo z

L4380

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4381

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4382

Hypothesis H6 : SNo (x + y)

L4383

Hypothesis H7 : SNo ((x + y) * z)

L4384

Hypothesis H8 : SNo (y * z)

L4385

Hypothesis H9 : SNo w

L4386

Hypothesis H10 : u ∈ SNoL x

L4387

Hypothesis H11 : v ∈ SNoL z

L4388

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4389

Hypothesis H13 : SNo u

L4390

Hypothesis H14 : u < x

L4391

Hypothesis H15 : SNo v

L4392

Hypothesis H16 : v < z

L4393

Hypothesis H17 : SNo (u * v)

L4394

Hypothesis H18 : SNo (u + y)

L4395

Theorem. (Conj_mul_SNo_distrR__74__5)

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

In Proofgold the corresponding term root is 16c4be... and proposition id is d855fb...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__75__5

L4401

Variable x : set

(*** Conj_mul_SNo_distrR__75__5 TMTkW6kKiGcqjymbaJ4m2zpKnt97V3XRjj5 bounty of about 25 bars ***)

L4402

Variable y : set

L4403

Variable z : set

L4404

Variable w : set

L4405

Variable u : set

L4406

Variable v : set

L4407

Hypothesis H0 : SNo x

L4408

Hypothesis H1 : SNo y

L4409

Hypothesis H2 : SNo z

L4410

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4411

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4412

Hypothesis H6 : SNo (x + y)

L4413

Hypothesis H7 : SNo ((x + y) * z)

L4414

Hypothesis H8 : SNo (y * z)

L4415

Hypothesis H9 : SNo w

L4416

Hypothesis H10 : u ∈ SNoL x

L4417

Hypothesis H11 : v ∈ SNoL z

L4418

Hypothesis H12 : (w + u * v) ≤ u * z + x * v

L4419

Hypothesis H13 : SNo u

L4420

Hypothesis H14 : u < x

L4421

Hypothesis H15 : SNo v

L4422

Hypothesis H16 : v < z

L4423

Hypothesis H17 : SNo (u * v)

L4424

Theorem. (Conj_mul_SNo_distrR__75__5)

SNo (u + y) → (w + y * z) < (x + y) * z

In Proofgold the corresponding term root is 61b5d1... and proposition id is f92793...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__76__12

L4430

Variable x : set

(*** Conj_mul_SNo_distrR__76__12 TMYAEiux2ftJmL7dYKfKyomf2jQH1YeQ5M9 bounty of about 25 bars ***)

L4431

Variable y : set

L4432

Variable z : set

L4433

Variable w : set

L4434

Variable u : set

L4435

Variable v : set

L4436

Hypothesis H0 : SNo x

L4437

Hypothesis H1 : SNo y

L4438

Hypothesis H2 : SNo z

L4439

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4440

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + y) * x2 = x * x2 + y * x2)

L4441

Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4442

Hypothesis H6 : SNo (x + y)

L4443

Hypothesis H7 : SNo ((x + y) * z)

L4444

Hypothesis H8 : SNo (y * z)

L4445

Hypothesis H9 : SNo w

L4446

Hypothesis H10 : u ∈ SNoL x

L4447

Hypothesis H11 : v ∈ SNoL z

L4448

Hypothesis H13 : SNo u

L4449

Hypothesis H14 : u < x

L4450

Hypothesis H15 : SNo v

L4451

Hypothesis H16 : v < z

L4452

Theorem. (Conj_mul_SNo_distrR__76__12)

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

In Proofgold the corresponding term root is 2efaf1... and proposition id is d93f00...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__77__2

L4458

Variable x : set

(*** Conj_mul_SNo_distrR__77__2 TMdugbRqF1AudSCUMjJidHjDUPf77W8phGT bounty of about 25 bars ***)

L4459

Variable y : set

L4460

Variable z : set

L4461

Variable w : set

L4462

Variable u : set

L4463

Variable v : set

L4464

Hypothesis H0 : SNo x

L4465

Hypothesis H1 : SNo y

L4466

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L4467

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L4468

Hypothesis H5 : SNo (x * z)

L4469

Hypothesis H6 : SNo (y * z)

L4470

Hypothesis H7 : u ∈ SNoL z

L4471

Hypothesis H8 : SNo w

L4472

Hypothesis H9 : SNo u

L4473

Hypothesis H10 : u < z

L4474

Hypothesis H11 : SNo (x * u)

L4475

Hypothesis H12 : SNo (y * u)

L4476

Hypothesis H13 : SNo (w * z)

L4477

Hypothesis H14 : SNo (w * u)

L4478

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4479

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4480

Hypothesis H17 : v ∈ SNoR y

L4481

Hypothesis H18 : (x + v) ≤ w

L4482

Hypothesis H19 : SNo v

L4483

Hypothesis H20 : y < v

L4484

Hypothesis H21 : SNo (v * u)

L4485

Hypothesis H22 : SNo (v * z)

L4486

Theorem. (Conj_mul_SNo_distrR__77__2)

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

In Proofgold the corresponding term root is 2acdb5... and proposition id is 4d4193...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__77__13

L4492

Variable x : set

(*** Conj_mul_SNo_distrR__77__13 TMWkauJuAqaYcbJRnnZxFpaneBQ73EunD8T bounty of about 25 bars ***)

L4493

Variable y : set

L4494

Variable z : set

L4495

Variable w : set

L4496

Variable u : set

L4497

Variable v : set

L4498

Hypothesis H0 : SNo x

L4499

Hypothesis H1 : SNo y

L4500

Hypothesis H2 : SNo z

L4501

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L4502

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L4503

Hypothesis H5 : SNo (x * z)

L4504

Hypothesis H6 : SNo (y * z)

L4505

Hypothesis H7 : u ∈ SNoL z

L4506

Hypothesis H8 : SNo w

L4507

Hypothesis H9 : SNo u

L4508

Hypothesis H10 : u < z

L4509

Hypothesis H11 : SNo (x * u)

L4510

Hypothesis H12 : SNo (y * u)

L4511

Hypothesis H14 : SNo (w * u)

L4512

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4513

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4514

Hypothesis H17 : v ∈ SNoR y

L4515

Hypothesis H18 : (x + v) ≤ w

L4516

Hypothesis H19 : SNo v

L4517

Hypothesis H20 : y < v

L4518

Hypothesis H21 : SNo (v * u)

L4519

Hypothesis H22 : SNo (v * z)

L4520

Theorem. (Conj_mul_SNo_distrR__77__13)

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

In Proofgold the corresponding term root is 363b0f... and proposition id is 44a2ee...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__78__18

L4526

Variable x : set

(*** Conj_mul_SNo_distrR__78__18 TMVLi79f5ooai9sgjMPsN3EdCczK5LExmT7 bounty of about 25 bars ***)

L4527

Variable y : set

L4528

Variable z : set

L4529

Variable w : set

L4530

Variable u : set

L4531

Variable v : set

L4532

Hypothesis H0 : SNo x

L4533

Hypothesis H1 : SNo y

L4534

Hypothesis H2 : SNo z

L4535

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L4536

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L4537

Hypothesis H5 : SNo (x * z)

L4538

Hypothesis H6 : SNo (y * z)

L4539

Hypothesis H7 : u ∈ SNoL z

L4540

Hypothesis H8 : SNo w

L4541

Hypothesis H9 : SNo u

L4542

Hypothesis H10 : u < z

L4543

Hypothesis H11 : SNo (x * u)

L4544

Hypothesis H12 : SNo (y * u)

L4545

Hypothesis H13 : SNo (w * z)

L4546

Hypothesis H14 : SNo (w * u)

L4547

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4548

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4549

Hypothesis H17 : v ∈ SNoR y

L4550

Hypothesis H19 : SNo v

L4551

Hypothesis H20 : y < v

L4552

Hypothesis H21 : SNo (v * u)

L4553

Theorem. (Conj_mul_SNo_distrR__78__18)

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

In Proofgold the corresponding term root is 91ca30... and proposition id is 5a1230...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__80__0

L4559

Variable x : set

(*** Conj_mul_SNo_distrR__80__0 TMR57Nz5WAEUXE817U3ugbZ9v8kRLbVgnEH bounty of about 25 bars ***)

L4560

Variable y : set

L4561

Variable z : set

L4562

Variable w : set

L4563

Variable u : set

L4564

Variable v : set

L4565

Hypothesis H1 : SNo y

L4566

Hypothesis H2 : SNo z

L4567

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4568

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4569

Hypothesis H5 : SNo (x * z)

L4570

Hypothesis H6 : SNo (y * z)

L4571

Hypothesis H7 : u ∈ SNoL z

L4572

Hypothesis H8 : SNo w

L4573

Hypothesis H9 : SNo u

L4574

Hypothesis H10 : u < z

L4575

Hypothesis H11 : SNo (x * u)

L4576

Hypothesis H12 : SNo (y * u)

L4577

Hypothesis H13 : SNo (w * z)

L4578

Hypothesis H14 : SNo (w * u)

L4579

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4580

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4581

Hypothesis H17 : v ∈ SNoR x

L4582

Hypothesis H18 : (v + y) ≤ w

L4583

Hypothesis H19 : SNo v

L4584

Hypothesis H20 : x < v

L4585

Hypothesis H21 : SNo (v * u)

L4586

Hypothesis H22 : SNo (v * z)

L4587

Theorem. (Conj_mul_SNo_distrR__80__0)

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

In Proofgold the corresponding term root is 5d7bad... and proposition id is 026e39...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__80__7

L4593

Variable x : set

(*** Conj_mul_SNo_distrR__80__7 TMT8JVYbXfJ92euamdyePXNdrjYuu9gqm4z bounty of about 25 bars ***)

L4594

Variable y : set

L4595

Variable z : set

L4596

Variable w : set

L4597

Variable u : set

L4598

Variable v : set

L4599

Hypothesis H0 : SNo x

L4600

Hypothesis H1 : SNo y

L4601

Hypothesis H2 : SNo z

L4602

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4603

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4604

Hypothesis H5 : SNo (x * z)

L4605

Hypothesis H6 : SNo (y * z)

L4606

Hypothesis H8 : SNo w

L4607

Hypothesis H9 : SNo u

L4608

Hypothesis H10 : u < z

L4609

Hypothesis H11 : SNo (x * u)

L4610

Hypothesis H12 : SNo (y * u)

L4611

Hypothesis H13 : SNo (w * z)

L4612

Hypothesis H14 : SNo (w * u)

L4613

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4614

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4615

Hypothesis H17 : v ∈ SNoR x

L4616

Hypothesis H18 : (v + y) ≤ w

L4617

Hypothesis H19 : SNo v

L4618

Hypothesis H20 : x < v

L4619

Hypothesis H21 : SNo (v * u)

L4620

Hypothesis H22 : SNo (v * z)

L4621

Theorem. (Conj_mul_SNo_distrR__80__7)

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

In Proofgold the corresponding term root is cb8a21... and proposition id is b6bb04...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__80__13

L4627

Variable x : set

(*** Conj_mul_SNo_distrR__80__13 TMJ7u36B2TPRL7SyeEphjUU4sA5AmYomqG1 bounty of about 25 bars ***)

L4628

Variable y : set

L4629

Variable z : set

L4630

Variable w : set

L4631

Variable u : set

L4632

Variable v : set

L4633

Hypothesis H0 : SNo x

L4634

Hypothesis H1 : SNo y

L4635

Hypothesis H2 : SNo z

L4636

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4637

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4638

Hypothesis H5 : SNo (x * z)

L4639

Hypothesis H6 : SNo (y * z)

L4640

Hypothesis H7 : u ∈ SNoL z

L4641

Hypothesis H8 : SNo w

L4642

Hypothesis H9 : SNo u

L4643

Hypothesis H10 : u < z

L4644

Hypothesis H11 : SNo (x * u)

L4645

Hypothesis H12 : SNo (y * u)

L4646

Hypothesis H14 : SNo (w * u)

L4647

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4648

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4649

Hypothesis H17 : v ∈ SNoR x

L4650

Hypothesis H18 : (v + y) ≤ w

L4651

Hypothesis H19 : SNo v

L4652

Hypothesis H20 : x < v

L4653

Hypothesis H21 : SNo (v * u)

L4654

Hypothesis H22 : SNo (v * z)

L4655

Theorem. (Conj_mul_SNo_distrR__80__13)

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

In Proofgold the corresponding term root is 4d1def... and proposition id is 41789d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__80__16

L4661

Variable x : set

(*** Conj_mul_SNo_distrR__80__16 TMGgbV6t1hxLQCKqyRuWqRAx58k3ngxZifK bounty of about 25 bars ***)

L4662

Variable y : set

L4663

Variable z : set

L4664

Variable w : set

L4665

Variable u : set

L4666

Variable v : set

L4667

Hypothesis H0 : SNo x

L4668

Hypothesis H1 : SNo y

L4669

Hypothesis H2 : SNo z

L4670

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4671

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4672

Hypothesis H5 : SNo (x * z)

L4673

Hypothesis H6 : SNo (y * z)

L4674

Hypothesis H7 : u ∈ SNoL z

L4675

Hypothesis H8 : SNo w

L4676

Hypothesis H9 : SNo u

L4677

Hypothesis H10 : u < z

L4678

Hypothesis H11 : SNo (x * u)

L4679

Hypothesis H12 : SNo (y * u)

L4680

Hypothesis H13 : SNo (w * z)

L4681

Hypothesis H14 : SNo (w * u)

L4682

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4683

Hypothesis H17 : v ∈ SNoR x

L4684

Hypothesis H18 : (v + y) ≤ w

L4685

Hypothesis H19 : SNo v

L4686

Hypothesis H20 : x < v

L4687

Hypothesis H21 : SNo (v * u)

L4688

Hypothesis H22 : SNo (v * z)

L4689

Theorem. (Conj_mul_SNo_distrR__80__16)

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

In Proofgold the corresponding term root is 4163eb... and proposition id is 113576...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__81__7

L4695

Variable x : set

(*** Conj_mul_SNo_distrR__81__7 TMY3Cs5bDWCjKrHiSaH5ii4RqFKFQ2VyHvh bounty of about 25 bars ***)

L4696

Variable y : set

L4697

Variable z : set

L4698

Variable w : set

L4699

Variable u : set

L4700

Variable v : set

L4701

Hypothesis H0 : SNo x

L4702

Hypothesis H1 : SNo y

L4703

Hypothesis H2 : SNo z

L4704

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4705

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4706

Hypothesis H5 : SNo (x * z)

L4707

Hypothesis H6 : SNo (y * z)

L4708

Hypothesis H8 : SNo w

L4709

Hypothesis H9 : SNo u

L4710

Hypothesis H10 : u < z

L4711

Hypothesis H11 : SNo (x * u)

L4712

Hypothesis H12 : SNo (y * u)

L4713

Hypothesis H13 : SNo (w * z)

L4714

Hypothesis H14 : SNo (w * u)

L4715

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4716

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4717

Hypothesis H17 : v ∈ SNoR x

L4718

Hypothesis H18 : (v + y) ≤ w

L4719

Hypothesis H19 : SNo v

L4720

Hypothesis H20 : x < v

L4721

Hypothesis H21 : SNo (v * u)

L4722

Theorem. (Conj_mul_SNo_distrR__81__7)

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

In Proofgold the corresponding term root is a8c4a6... and proposition id is 750580...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__81__9

L4728

Variable x : set

(*** Conj_mul_SNo_distrR__81__9 TMNaP1cQ9ixnsnbZvwm1g5yFAY7SaNddXc5 bounty of about 25 bars ***)

L4729

Variable y : set

L4730

Variable z : set

L4731

Variable w : set

L4732

Variable u : set

L4733

Variable v : set

L4734

Hypothesis H0 : SNo x

L4735

Hypothesis H1 : SNo y

L4736

Hypothesis H2 : SNo z

L4737

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4738

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4739

Hypothesis H5 : SNo (x * z)

L4740

Hypothesis H6 : SNo (y * z)

L4741

Hypothesis H7 : u ∈ SNoL z

L4742

Hypothesis H8 : SNo w

L4743

Hypothesis H10 : u < z

L4744

Hypothesis H11 : SNo (x * u)

L4745

Hypothesis H12 : SNo (y * u)

L4746

Hypothesis H13 : SNo (w * z)

L4747

Hypothesis H14 : SNo (w * u)

L4748

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4749

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4750

Hypothesis H17 : v ∈ SNoR x

L4751

Hypothesis H18 : (v + y) ≤ w

L4752

Hypothesis H19 : SNo v

L4753

Hypothesis H20 : x < v

L4754

Hypothesis H21 : SNo (v * u)

L4755

Theorem. (Conj_mul_SNo_distrR__81__9)

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

In Proofgold the corresponding term root is ba386e... and proposition id is b3b543...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__81__10

L4761

Variable x : set

(*** Conj_mul_SNo_distrR__81__10 TMShdL6GozWZ1axGsoLsssjmXpiE7nbxhnQ bounty of about 25 bars ***)

L4762

Variable y : set

L4763

Variable z : set

L4764

Variable w : set

L4765

Variable u : set

L4766

Variable v : set

L4767

Hypothesis H0 : SNo x

L4768

Hypothesis H1 : SNo y

L4769

Hypothesis H2 : SNo z

L4770

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4771

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4772

Hypothesis H5 : SNo (x * z)

L4773

Hypothesis H6 : SNo (y * z)

L4774

Hypothesis H7 : u ∈ SNoL z

L4775

Hypothesis H8 : SNo w

L4776

Hypothesis H9 : SNo u

L4777

Hypothesis H11 : SNo (x * u)

L4778

Hypothesis H12 : SNo (y * u)

L4779

Hypothesis H13 : SNo (w * z)

L4780

Hypothesis H14 : SNo (w * u)

L4781

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4782

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4783

Hypothesis H17 : v ∈ SNoR x

L4784

Hypothesis H18 : (v + y) ≤ w

L4785

Hypothesis H19 : SNo v

L4786

Hypothesis H20 : x < v

L4787

Hypothesis H21 : SNo (v * u)

L4788

Theorem. (Conj_mul_SNo_distrR__81__10)

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

In Proofgold the corresponding term root is 2cb931... and proposition id is 282fd4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__81__19

L4794

Variable x : set

(*** Conj_mul_SNo_distrR__81__19 TMJh7xa5c7G9F29UBVvcWDGrnT43zspkkHL bounty of about 25 bars ***)

L4795

Variable y : set

L4796

Variable z : set

L4797

Variable w : set

L4798

Variable u : set

L4799

Variable v : set

L4800

Hypothesis H0 : SNo x

L4801

Hypothesis H1 : SNo y

L4802

Hypothesis H2 : SNo z

L4803

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4804

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4805

Hypothesis H5 : SNo (x * z)

L4806

Hypothesis H6 : SNo (y * z)

L4807

Hypothesis H7 : u ∈ SNoL z

L4808

Hypothesis H8 : SNo w

L4809

Hypothesis H9 : SNo u

L4810

Hypothesis H10 : u < z

L4811

Hypothesis H11 : SNo (x * u)

L4812

Hypothesis H12 : SNo (y * u)

L4813

Hypothesis H13 : SNo (w * z)

L4814

Hypothesis H14 : SNo (w * u)

L4815

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4816

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4817

Hypothesis H17 : v ∈ SNoR x

L4818

Hypothesis H18 : (v + y) ≤ w

L4819

Hypothesis H20 : x < v

L4820

Hypothesis H21 : SNo (v * u)

L4821

Theorem. (Conj_mul_SNo_distrR__81__19)

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

In Proofgold the corresponding term root is 8c8f28... and proposition id is 53f602...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__82__13

L4827

Variable x : set

(*** Conj_mul_SNo_distrR__82__13 TMbmp3VPbw8DP8kVyEkDGEWq37HxkvmKF7v bounty of about 25 bars ***)

L4828

Variable y : set

L4829

Variable z : set

L4830

Variable w : set

L4831

Variable u : set

L4832

Variable v : set

L4833

Hypothesis H0 : SNo x

L4834

Hypothesis H1 : SNo y

L4835

Hypothesis H2 : SNo z

L4836

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L4837

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L4838

Hypothesis H5 : SNo (x * z)

L4839

Hypothesis H6 : SNo (y * z)

L4840

Hypothesis H7 : u ∈ SNoL z

L4841

Hypothesis H8 : SNo w

L4842

Hypothesis H9 : SNo u

L4843

Hypothesis H10 : u < z

L4844

Hypothesis H11 : SNo (x * u)

L4845

Hypothesis H12 : SNo (y * u)

L4846

Hypothesis H14 : SNo (w * u)

L4847

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4848

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4849

Hypothesis H17 : v ∈ SNoR x

L4850

Hypothesis H18 : (v + y) ≤ w

L4851

Hypothesis H19 : SNo v

L4852

Hypothesis H20 : x < v

L4853

Theorem. (Conj_mul_SNo_distrR__82__13)

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

In Proofgold the corresponding term root is 4008b6... and proposition id is fc50dc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__83__8

L4859

Variable x : set

(*** Conj_mul_SNo_distrR__83__8 TMGvJMkcjaxY8pSmEjav1UrPSUHkPV9CHrp bounty of about 25 bars ***)

L4860

Variable y : set

L4861

Variable z : set

L4862

Variable w : set

L4863

Variable u : set

L4864

Hypothesis H0 : SNo x

L4865

Hypothesis H1 : SNo y

L4866

Hypothesis H2 : SNo z

L4867

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L4868

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L4869

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L4870

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L4871

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L4872

Hypothesis H9 : SNo (y * z)

L4873

Hypothesis H10 : SNo (x * z + y * z)

L4874

Hypothesis H11 : w ∈ SNoR (x + y)

L4875

Hypothesis H12 : u ∈ SNoL z

L4876

Hypothesis H13 : SNo w

L4877

Hypothesis H14 : SNo u

L4878

Hypothesis H15 : u < z

L4879

Hypothesis H16 : SNo (x * u)

L4880

Hypothesis H17 : SNo (y * u)

L4881

Hypothesis H18 : SNo (w * z)

L4882

Hypothesis H19 : SNo ((x + y) * u)

L4883

Hypothesis H20 : SNo (w * u)

L4884

Hypothesis H21 : SNo (w * z + x * u + y * u)

L4885

Theorem. (Conj_mul_SNo_distrR__83__8)

SNo (x * z + y * z + w * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 2bc400... and proposition id is 158669...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__85__18

L4891

Variable x : set

(*** Conj_mul_SNo_distrR__85__18 TMPU9rb2gCEPYzveAHTBZntzdbVWWGxPREr bounty of about 25 bars ***)

L4892

Variable y : set

L4893

Variable z : set

L4894

Variable w : set

L4895

Variable u : set

L4896

Hypothesis H0 : SNo x

L4897

Hypothesis H1 : SNo y

L4898

Hypothesis H2 : SNo z

L4899

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L4900

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L4901

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L4902

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L4903

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L4904

Hypothesis H8 : SNo (x * z)

L4905

Hypothesis H9 : SNo (y * z)

L4906

Hypothesis H10 : SNo (x * z + y * z)

L4907

Hypothesis H11 : w ∈ SNoR (x + y)

L4908

Hypothesis H12 : u ∈ SNoL z

L4909

Hypothesis H13 : SNo w

L4910

Hypothesis H14 : SNo u

L4911

Hypothesis H15 : u < z

L4912

Hypothesis H16 : SNo (x * u)

L4913

Hypothesis H17 : SNo (y * u)

L4914

Hypothesis H19 : SNo ((x + y) * u)

L4915

Theorem. (Conj_mul_SNo_distrR__85__18)

SNo (w * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 66e6ab... and proposition id is 5fda55...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__90__15

L4921

Variable x : set

(*** Conj_mul_SNo_distrR__90__15 TMVZSi6EwpMF7T395XzsM67rQ2ufThh2r4E bounty of about 25 bars ***)

L4922

Variable y : set

L4923

Variable z : set

L4924

Variable w : set

L4925

Variable u : set

L4926

Hypothesis H0 : SNo x

L4927

Hypothesis H1 : SNo y

L4928

Hypothesis H2 : SNo z

L4929

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L4930

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L4931

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L4932

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L4933

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L4934

Hypothesis H8 : SNo (x + y)

L4935

Hypothesis H9 : SNo (x * z)

L4936

Hypothesis H10 : SNo (y * z)

L4937

Hypothesis H11 : SNo (x * z + y * z)

L4938

Hypothesis H12 : w ∈ SNoR (x + y)

L4939

Hypothesis H13 : u ∈ SNoL z

L4940

Hypothesis H14 : SNo w

L4941

Hypothesis H16 : u < z

L4942

Theorem. (Conj_mul_SNo_distrR__90__15)

SNo (x * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is f9425e... and proposition id is e20d76...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__92__5

L4948

Variable x : set

(*** Conj_mul_SNo_distrR__92__5 TMZJ3ghx23sYEWD8VhXCqKwaY6PXbhLzC4N bounty of about 25 bars ***)

L4949

Variable y : set

L4950

Variable z : set

L4951

Variable w : set

L4952

Variable u : set

L4953

Variable v : set

L4954

Hypothesis H0 : SNo x

L4955

Hypothesis H1 : SNo y

L4956

Hypothesis H2 : SNo z

L4957

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L4958

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L4959

Hypothesis H6 : SNo (y * z)

L4960

Hypothesis H7 : u ∈ SNoR z

L4961

Hypothesis H8 : SNo w

L4962

Hypothesis H9 : SNo u

L4963

Hypothesis H10 : z < u

L4964

Hypothesis H11 : SNo (x * u)

L4965

Hypothesis H12 : SNo (y * u)

L4966

Hypothesis H13 : SNo (w * z)

L4967

Hypothesis H14 : SNo (w * u)

L4968

Hypothesis H15 : SNo (w * z + x * u + y * u)

L4969

Hypothesis H16 : SNo (x * z + y * z + w * u)

L4970

Hypothesis H17 : v ∈ SNoL y

L4971

Hypothesis H18 : w ≤ x + v

L4972

Hypothesis H19 : SNo v

L4973

Hypothesis H20 : v < y

L4974

Hypothesis H21 : SNo (v * u)

L4975

Theorem. (Conj_mul_SNo_distrR__92__5)

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

In Proofgold the corresponding term root is cf93a2... and proposition id is dd9029...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__92__16

L4981

Variable x : set

(*** Conj_mul_SNo_distrR__92__16 TMYd1RPzXPRBXSo7U5KAEoDNf1UdDXteTnq bounty of about 25 bars ***)

L4982

Variable y : set

L4983

Variable z : set

L4984

Variable w : set

L4985

Variable u : set

L4986

Variable v : set

L4987

Hypothesis H0 : SNo x

L4988

Hypothesis H1 : SNo y

L4989

Hypothesis H2 : SNo z

L4990

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L4991

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L4992

Hypothesis H5 : SNo (x * z)

L4993

Hypothesis H6 : SNo (y * z)

L4994

Hypothesis H7 : u ∈ SNoR z

L4995

Hypothesis H8 : SNo w

L4996

Hypothesis H9 : SNo u

L4997

Hypothesis H10 : z < u

L4998

Hypothesis H11 : SNo (x * u)

L4999

Hypothesis H12 : SNo (y * u)

L5000

Hypothesis H13 : SNo (w * z)

L5001

Hypothesis H14 : SNo (w * u)

L5002

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5003

Hypothesis H17 : v ∈ SNoL y

L5004

Hypothesis H18 : w ≤ x + v

L5005

Hypothesis H19 : SNo v

L5006

Hypothesis H20 : v < y

L5007

Hypothesis H21 : SNo (v * u)

L5008

Theorem. (Conj_mul_SNo_distrR__92__16)

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

In Proofgold the corresponding term root is a874e3... and proposition id is 23c479...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__93__0

L5014

Variable x : set

(*** Conj_mul_SNo_distrR__93__0 TMV3sKQA4Sh62SQtRQFqUh2RYFXKpksXJdv bounty of about 25 bars ***)

L5015

Variable y : set

L5016

Variable z : set

L5017

Variable w : set

L5018

Variable u : set

L5019

Variable v : set

L5020

Hypothesis H1 : SNo y

L5021

Hypothesis H2 : SNo z

L5022

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L5023

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L5024

Hypothesis H5 : SNo (x * z)

L5025

Hypothesis H6 : SNo (y * z)

L5026

Hypothesis H7 : u ∈ SNoR z

L5027

Hypothesis H8 : SNo w

L5028

Hypothesis H9 : SNo u

L5029

Hypothesis H10 : z < u

L5030

Hypothesis H11 : SNo (x * u)

L5031

Hypothesis H12 : SNo (y * u)

L5032

Hypothesis H13 : SNo (w * z)

L5033

Hypothesis H14 : SNo (w * u)

L5034

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5035

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5036

Hypothesis H17 : v ∈ SNoL y

L5037

Hypothesis H18 : w ≤ x + v

L5038

Hypothesis H19 : SNo v

L5039

Hypothesis H20 : v < y

L5040

Theorem. (Conj_mul_SNo_distrR__93__0)

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

In Proofgold the corresponding term root is 2a404b... and proposition id is 26c987...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__94__0

L5046

Variable x : set

(*** Conj_mul_SNo_distrR__94__0 TMW5J1p6ri2h3XpxHqA7qdGhqoN4rd1Wmyx bounty of about 25 bars ***)

L5047

Variable y : set

L5048

Variable z : set

L5049

Variable w : set

L5050

Variable u : set

L5051

Variable v : set

L5052

Hypothesis H1 : SNo y

L5053

Hypothesis H2 : SNo z

L5054

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5055

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5056

Hypothesis H5 : SNo (x * z)

L5057

Hypothesis H6 : SNo (y * z)

L5058

Hypothesis H7 : u ∈ SNoR z

L5059

Hypothesis H8 : SNo w

L5060

Hypothesis H9 : SNo u

L5061

Hypothesis H10 : z < u

L5062

Hypothesis H11 : SNo (x * u)

L5063

Hypothesis H12 : SNo (y * u)

L5064

Hypothesis H13 : SNo (w * z)

L5065

Hypothesis H14 : SNo (w * u)

L5066

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5067

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5068

Hypothesis H17 : v ∈ SNoL x

L5069

Hypothesis H18 : w ≤ v + y

L5070

Hypothesis H19 : SNo v

L5071

Hypothesis H20 : v < x

L5072

Hypothesis H21 : SNo (v * u)

L5073

Hypothesis H22 : SNo (v * z)

L5074

Theorem. (Conj_mul_SNo_distrR__94__0)

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

In Proofgold the corresponding term root is 23142b... and proposition id is e08a10...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__94__22

L5080

Variable x : set

(*** Conj_mul_SNo_distrR__94__22 TMGVpJbjSZw11rfp9Q8v3zkAbYkVECD3HjL bounty of about 25 bars ***)

L5081

Variable y : set

L5082

Variable z : set

L5083

Variable w : set

L5084

Variable u : set

L5085

Variable v : set

L5086

Hypothesis H0 : SNo x

L5087

Hypothesis H1 : SNo y

L5088

Hypothesis H2 : SNo z

L5089

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5090

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5091

Hypothesis H5 : SNo (x * z)

L5092

Hypothesis H6 : SNo (y * z)

L5093

Hypothesis H7 : u ∈ SNoR z

L5094

Hypothesis H8 : SNo w

L5095

Hypothesis H9 : SNo u

L5096

Hypothesis H10 : z < u

L5097

Hypothesis H11 : SNo (x * u)

L5098

Hypothesis H12 : SNo (y * u)

L5099

Hypothesis H13 : SNo (w * z)

L5100

Hypothesis H14 : SNo (w * u)

L5101

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5102

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5103

Hypothesis H17 : v ∈ SNoL x

L5104

Hypothesis H18 : w ≤ v + y

L5105

Hypothesis H19 : SNo v

L5106

Hypothesis H20 : v < x

L5107

Hypothesis H21 : SNo (v * u)

L5108

Theorem. (Conj_mul_SNo_distrR__94__22)

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

In Proofgold the corresponding term root is 6b8aaa... and proposition id is 68dcca...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__95__12

L5114

Variable x : set

(*** Conj_mul_SNo_distrR__95__12 TMTjcbr4qgJB2j2bzwH3jJDPw7HLTc5kjzi bounty of about 25 bars ***)

L5115

Variable y : set

L5116

Variable z : set

L5117

Variable w : set

L5118

Variable u : set

L5119

Variable v : set

L5120

Hypothesis H0 : SNo x

L5121

Hypothesis H1 : SNo y

L5122

Hypothesis H2 : SNo z

L5123

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5124

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5125

Hypothesis H5 : SNo (x * z)

L5126

Hypothesis H6 : SNo (y * z)

L5127

Hypothesis H7 : u ∈ SNoR z

L5128

Hypothesis H8 : SNo w

L5129

Hypothesis H9 : SNo u

L5130

Hypothesis H10 : z < u

L5131

Hypothesis H11 : SNo (x * u)

L5132

Hypothesis H13 : SNo (w * z)

L5133

Hypothesis H14 : SNo (w * u)

L5134

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5135

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5136

Hypothesis H17 : v ∈ SNoL x

L5137

Hypothesis H18 : w ≤ v + y

L5138

Hypothesis H19 : SNo v

L5139

Hypothesis H20 : v < x

L5140

Hypothesis H21 : SNo (v * u)

L5141

Theorem. (Conj_mul_SNo_distrR__95__12)

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

In Proofgold the corresponding term root is 78d906... and proposition id is be0e9a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__97__18

L5147

Variable x : set

(*** Conj_mul_SNo_distrR__97__18 TMYJy1GcCAhUHPjV5HDcGoCAwdrxkaMjvcm bounty of about 25 bars ***)

L5148

Variable y : set

L5149

Variable z : set

L5150

Variable w : set

L5151

Variable u : set

L5152

Hypothesis H0 : SNo x

L5153

Hypothesis H1 : SNo y

L5154

Hypothesis H2 : SNo z

L5155

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5156

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5157

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5158

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5159

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5160

Hypothesis H8 : SNo (x * z)

L5161

Hypothesis H9 : SNo (y * z)

L5162

Hypothesis H10 : SNo (x * z + y * z)

L5163

Hypothesis H11 : w ∈ SNoL (x + y)

L5164

Hypothesis H12 : u ∈ SNoR z

L5165

Hypothesis H13 : SNo w

L5166

Hypothesis H14 : SNo u

L5167

Hypothesis H15 : z < u

L5168

Hypothesis H16 : SNo (x * u)

L5169

Hypothesis H17 : SNo (y * u)

L5170

Hypothesis H19 : SNo ((x + y) * u)

L5171

Hypothesis H20 : SNo (w * u)

L5172

Hypothesis H21 : SNo (w * z + x * u + y * u)

L5173

Theorem. (Conj_mul_SNo_distrR__97__18)

SNo (x * z + y * z + w * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 233d31... and proposition id is 3f1396...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__98__10

L5179

Variable x : set

(*** Conj_mul_SNo_distrR__98__10 TMT8cWBkCkTEcypsJn99yRjGXYLVgeCN3Cn bounty of about 25 bars ***)

L5180

Variable y : set

L5181

Variable z : set

L5182

Variable w : set

L5183

Variable u : set

L5184

Hypothesis H0 : SNo x

L5185

Hypothesis H1 : SNo y

L5186

Hypothesis H2 : SNo z

L5187

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5188

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5189

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5190

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5191

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5192

Hypothesis H8 : SNo (x * z)

L5193

Hypothesis H9 : SNo (y * z)

L5194

Hypothesis H11 : w ∈ SNoL (x + y)

L5195

Hypothesis H12 : u ∈ SNoR z

L5196

Hypothesis H13 : SNo w

L5197

Hypothesis H14 : SNo u

L5198

Hypothesis H15 : z < u

L5199

Hypothesis H16 : SNo (x * u)

L5200

Hypothesis H17 : SNo (y * u)

L5201

Hypothesis H18 : SNo (w * z)

L5202

Hypothesis H19 : SNo ((x + y) * u)

L5203

Hypothesis H20 : SNo (w * u)

L5204

Theorem. (Conj_mul_SNo_distrR__98__10)

SNo (w * z + x * u + y * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 867458... and proposition id is 27c437...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__100__6

L5210

Variable x : set

(*** Conj_mul_SNo_distrR__100__6 TMYakRNoEJbjsTqCp4RqMrptQvBkJwwJpf1 bounty of about 25 bars ***)

L5211

Variable y : set

L5212

Variable z : set

L5213

Variable w : set

L5214

Variable u : set

L5215

Hypothesis H0 : SNo x

L5216

Hypothesis H1 : SNo y

L5217

Hypothesis H2 : SNo z

L5218

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5219

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5220

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5221

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5222

Hypothesis H8 : SNo (x * z)

L5223

Hypothesis H9 : SNo (y * z)

L5224

Hypothesis H10 : SNo (x * z + y * z)

L5225

Hypothesis H11 : w ∈ SNoL (x + y)

L5226

Hypothesis H12 : u ∈ SNoR z

L5227

Hypothesis H13 : SNo w

L5228

Hypothesis H14 : SNo u

L5229

Hypothesis H15 : z < u

L5230

Hypothesis H16 : SNo (x * u)

L5231

Hypothesis H17 : SNo (y * u)

L5232

Hypothesis H18 : SNo (w * z)

L5233

Hypothesis H19 : SNo ((x + y) * u)

L5234

Theorem. (Conj_mul_SNo_distrR__100__6)

SNo (x * u + y * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is a24b79... and proposition id is 36a85c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__100__13

L5240

Variable x : set

(*** Conj_mul_SNo_distrR__100__13 TMaqP72VD68s2zKHcaCDzWNCPhxoeC5Ukct bounty of about 25 bars ***)

L5241

Variable y : set

L5242

Variable z : set

L5243

Variable w : set

L5244

Variable u : set

L5245

Hypothesis H0 : SNo x

L5246

Hypothesis H1 : SNo y

L5247

Hypothesis H2 : SNo z

L5248

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5249

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5250

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5251

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5252

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5253

Hypothesis H8 : SNo (x * z)

L5254

Hypothesis H9 : SNo (y * z)

L5255

Hypothesis H10 : SNo (x * z + y * z)

L5256

Hypothesis H11 : w ∈ SNoL (x + y)

L5257

Hypothesis H12 : u ∈ SNoR z

L5258

Hypothesis H14 : SNo u

L5259

Hypothesis H15 : z < u

L5260

Hypothesis H16 : SNo (x * u)

L5261

Hypothesis H17 : SNo (y * u)

L5262

Hypothesis H18 : SNo (w * z)

L5263

Hypothesis H19 : SNo ((x + y) * u)

L5264

Theorem. (Conj_mul_SNo_distrR__100__13)

SNo (x * u + y * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 6c44aa... and proposition id is 662d2b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__101__14

L5270

Variable x : set

(*** Conj_mul_SNo_distrR__101__14 TMH8JteUWPM6shMFTW2Db98abzc3HtqCKnr bounty of about 25 bars ***)

L5271

Variable y : set

L5272

Variable z : set

L5273

Variable w : set

L5274

Variable u : set

L5275

Hypothesis H0 : SNo x

L5276

Hypothesis H1 : SNo y

L5277

Hypothesis H2 : SNo z

L5278

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5279

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5280

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5281

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5282

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5283

Hypothesis H8 : SNo (x + y)

L5284

Hypothesis H9 : SNo (x * z)

L5285

Hypothesis H10 : SNo (y * z)

L5286

Hypothesis H11 : SNo (x * z + y * z)

L5287

Hypothesis H12 : w ∈ SNoL (x + y)

L5288

Hypothesis H13 : u ∈ SNoR z

L5289

Hypothesis H15 : SNo u

L5290

Hypothesis H16 : z < u

L5291

Hypothesis H17 : SNo (x * u)

L5292

Hypothesis H18 : SNo (y * u)

L5293

Hypothesis H19 : SNo (w * z)

L5294

Theorem. (Conj_mul_SNo_distrR__101__14)

SNo ((x + y) * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 7a7f11... and proposition id is 99e27b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__104__2

L5300

Variable x : set

(*** Conj_mul_SNo_distrR__104__2 TMRrzkzxBWvj9eKcFmJAryE4BHsoL9JWxZ3 bounty of about 25 bars ***)

L5301

Variable y : set

L5302

Variable z : set

L5303

Variable w : set

L5304

Variable u : set

L5305

Hypothesis H0 : SNo x

L5306

Hypothesis H1 : SNo y

L5307

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5308

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5309

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5310

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5311

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5312

Hypothesis H8 : SNo (x + y)

L5313

Hypothesis H9 : SNo (x * z)

L5314

Hypothesis H10 : SNo (y * z)

L5315

Hypothesis H11 : SNo (x * z + y * z)

L5316

Hypothesis H12 : w ∈ SNoL (x + y)

L5317

Hypothesis H13 : u ∈ SNoR z

L5318

Hypothesis H14 : SNo w

L5319

Hypothesis H15 : SNo u

L5320

Hypothesis H16 : z < u

L5321

Theorem. (Conj_mul_SNo_distrR__104__2)

SNo (x * u) → (x * z + y * z) < w * z + (x + y) * u + - (w * u)

In Proofgold the corresponding term root is 041fbe... and proposition id is ba4879...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__105__19

L5327

Variable x : set

(*** Conj_mul_SNo_distrR__105__19 TMKSAH3dGmGsGNcFrm78y4Ft8chzCFwgCjN bounty of about 25 bars ***)

L5328

Variable y : set

L5329

Variable z : set

L5330

Variable w : set

L5331

Variable u : set

L5332

Variable v : set

L5333

Hypothesis H0 : SNo x

L5334

Hypothesis H1 : SNo y

L5335

Hypothesis H2 : SNo z

L5336

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L5337

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L5338

Hypothesis H5 : SNo (x * z)

L5339

Hypothesis H6 : SNo (y * z)

L5340

Hypothesis H7 : u ∈ SNoR z

L5341

Hypothesis H8 : SNo w

L5342

Hypothesis H9 : SNo u

L5343

Hypothesis H10 : z < u

L5344

Hypothesis H11 : SNo (x * u)

L5345

Hypothesis H12 : SNo (y * u)

L5346

Hypothesis H13 : SNo (w * z)

L5347

Hypothesis H14 : SNo (w * u)

L5348

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5349

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5350

Hypothesis H17 : v ∈ SNoR y

L5351

Hypothesis H18 : (x + v) ≤ w

L5352

Hypothesis H20 : y < v

L5353

Hypothesis H21 : SNo (v * u)

L5354

Hypothesis H22 : SNo (v * z)

L5355

Theorem. (Conj_mul_SNo_distrR__105__19)

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

In Proofgold the corresponding term root is 328cb5... and proposition id is a9bd4d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__107__18

L5361

Variable x : set

(*** Conj_mul_SNo_distrR__107__18 TMKmKcNTBSArFcrocLmff7btcAhdN8iBE8e bounty of about 25 bars ***)

L5362

Variable y : set

L5363

Variable z : set

L5364

Variable w : set

L5365

Variable u : set

L5366

Variable v : set

L5367

Hypothesis H0 : SNo x

L5368

Hypothesis H1 : SNo y

L5369

Hypothesis H2 : SNo z

L5370

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L5371

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L5372

Hypothesis H5 : SNo (x * z)

L5373

Hypothesis H6 : SNo (y * z)

L5374

Hypothesis H7 : u ∈ SNoR z

L5375

Hypothesis H8 : SNo w

L5376

Hypothesis H9 : SNo u

L5377

Hypothesis H10 : z < u

L5378

Hypothesis H11 : SNo (x * u)

L5379

Hypothesis H12 : SNo (y * u)

L5380

Hypothesis H13 : SNo (w * z)

L5381

Hypothesis H14 : SNo (w * u)

L5382

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5383

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5384

Hypothesis H17 : v ∈ SNoR y

L5385

Hypothesis H19 : SNo v

L5386

Hypothesis H20 : y < v

L5387

Theorem. (Conj_mul_SNo_distrR__107__18)

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

In Proofgold the corresponding term root is b08ec6... and proposition id is af4c67...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__108__2

L5393

Variable x : set

(*** Conj_mul_SNo_distrR__108__2 TMbxGoSFTWK8QHKDvsfq6e7TMu7eKLQpbpb bounty of about 25 bars ***)

L5394

Variable y : set

L5395

Variable z : set

L5396

Variable w : set

L5397

Variable u : set

L5398

Variable v : set

L5399

Hypothesis H0 : SNo x

L5400

Hypothesis H1 : SNo y

L5401

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5402

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5403

Hypothesis H5 : SNo (x * z)

L5404

Hypothesis H6 : SNo (y * z)

L5405

Hypothesis H7 : u ∈ SNoR z

L5406

Hypothesis H8 : SNo w

L5407

Hypothesis H9 : SNo u

L5408

Hypothesis H10 : z < u

L5409

Hypothesis H11 : SNo (x * u)

L5410

Hypothesis H12 : SNo (y * u)

L5411

Hypothesis H13 : SNo (w * z)

L5412

Hypothesis H14 : SNo (w * u)

L5413

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5414

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5415

Hypothesis H17 : v ∈ SNoR x

L5416

Hypothesis H18 : (v + y) ≤ w

L5417

Hypothesis H19 : SNo v

L5418

Hypothesis H20 : x < v

L5419

Hypothesis H21 : SNo (v * u)

L5420

Hypothesis H22 : SNo (v * z)

L5421

Theorem. (Conj_mul_SNo_distrR__108__2)

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

In Proofgold the corresponding term root is 9f7882... and proposition id is 76eb75...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__108__9

L5427

Variable x : set

(*** Conj_mul_SNo_distrR__108__9 TMct4mbr4YRkQEBrCxTtpC9uLWg5k3AkBst bounty of about 25 bars ***)

L5428

Variable y : set

L5429

Variable z : set

L5430

Variable w : set

L5431

Variable u : set

L5432

Variable v : set

L5433

Hypothesis H0 : SNo x

L5434

Hypothesis H1 : SNo y

L5435

Hypothesis H2 : SNo z

L5436

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5437

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5438

Hypothesis H5 : SNo (x * z)

L5439

Hypothesis H6 : SNo (y * z)

L5440

Hypothesis H7 : u ∈ SNoR z

L5441

Hypothesis H8 : SNo w

L5442

Hypothesis H10 : z < u

L5443

Hypothesis H11 : SNo (x * u)

L5444

Hypothesis H12 : SNo (y * u)

L5445

Hypothesis H13 : SNo (w * z)

L5446

Hypothesis H14 : SNo (w * u)

L5447

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5448

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5449

Hypothesis H17 : v ∈ SNoR x

L5450

Hypothesis H18 : (v + y) ≤ w

L5451

Hypothesis H19 : SNo v

L5452

Hypothesis H20 : x < v

L5453

Hypothesis H21 : SNo (v * u)

L5454

Hypothesis H22 : SNo (v * z)

L5455

Theorem. (Conj_mul_SNo_distrR__108__9)

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

In Proofgold the corresponding term root is 906a2c... and proposition id is 513e6e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__109__8

L5461

Variable x : set

(*** Conj_mul_SNo_distrR__109__8 TMR29mPKBrRdXxvXXa2ABtwUAtyYCgtQuKg bounty of about 25 bars ***)

L5462

Variable y : set

L5463

Variable z : set

L5464

Variable w : set

L5465

Variable u : set

L5466

Variable v : set

L5467

Hypothesis H0 : SNo x

L5468

Hypothesis H1 : SNo y

L5469

Hypothesis H2 : SNo z

L5470

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5471

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5472

Hypothesis H5 : SNo (x * z)

L5473

Hypothesis H6 : SNo (y * z)

L5474

Hypothesis H7 : u ∈ SNoR z

L5475

Hypothesis H9 : SNo u

L5476

Hypothesis H10 : z < u

L5477

Hypothesis H11 : SNo (x * u)

L5478

Hypothesis H12 : SNo (y * u)

L5479

Hypothesis H13 : SNo (w * z)

L5480

Hypothesis H14 : SNo (w * u)

L5481

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5482

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5483

Hypothesis H17 : v ∈ SNoR x

L5484

Hypothesis H18 : (v + y) ≤ w

L5485

Hypothesis H19 : SNo v

L5486

Hypothesis H20 : x < v

L5487

Hypothesis H21 : SNo (v * u)

L5488

Theorem. (Conj_mul_SNo_distrR__109__8)

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

In Proofgold the corresponding term root is 9263d6... and proposition id is aadce9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__109__18

L5494

Variable x : set

(*** Conj_mul_SNo_distrR__109__18 TMarrrQcBPDY9ANwKErTgSQcuXFFPeGPYyR bounty of about 25 bars ***)

L5495

Variable y : set

L5496

Variable z : set

L5497

Variable w : set

L5498

Variable u : set

L5499

Variable v : set

L5500

Hypothesis H0 : SNo x

L5501

Hypothesis H1 : SNo y

L5502

Hypothesis H2 : SNo z

L5503

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5504

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5505

Hypothesis H5 : SNo (x * z)

L5506

Hypothesis H6 : SNo (y * z)

L5507

Hypothesis H7 : u ∈ SNoR z

L5508

Hypothesis H8 : SNo w

L5509

Hypothesis H9 : SNo u

L5510

Hypothesis H10 : z < u

L5511

Hypothesis H11 : SNo (x * u)

L5512

Hypothesis H12 : SNo (y * u)

L5513

Hypothesis H13 : SNo (w * z)

L5514

Hypothesis H14 : SNo (w * u)

L5515

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5516

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5517

Hypothesis H17 : v ∈ SNoR x

L5518

Hypothesis H19 : SNo v

L5519

Hypothesis H20 : x < v

L5520

Hypothesis H21 : SNo (v * u)

L5521

Theorem. (Conj_mul_SNo_distrR__109__18)

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

In Proofgold the corresponding term root is 1bc39f... and proposition id is 4d5cb2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__110__16

L5527

Variable x : set

(*** Conj_mul_SNo_distrR__110__16 TMT3SmXYNFrPGLCpf1JXtoh6fuYCoBBXhsj bounty of about 25 bars ***)

L5528

Variable y : set

L5529

Variable z : set

L5530

Variable w : set

L5531

Variable u : set

L5532

Variable v : set

L5533

Hypothesis H0 : SNo x

L5534

Hypothesis H1 : SNo y

L5535

Hypothesis H2 : SNo z

L5536

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5537

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5538

Hypothesis H5 : SNo (x * z)

L5539

Hypothesis H6 : SNo (y * z)

L5540

Hypothesis H7 : u ∈ SNoR z

L5541

Hypothesis H8 : SNo w

L5542

Hypothesis H9 : SNo u

L5543

Hypothesis H10 : z < u

L5544

Hypothesis H11 : SNo (x * u)

L5545

Hypothesis H12 : SNo (y * u)

L5546

Hypothesis H13 : SNo (w * z)

L5547

Hypothesis H14 : SNo (w * u)

L5548

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5549

Hypothesis H17 : v ∈ SNoR x

L5550

Hypothesis H18 : (v + y) ≤ w

L5551

Hypothesis H19 : SNo v

L5552

Hypothesis H20 : x < v

L5553

Theorem. (Conj_mul_SNo_distrR__110__16)

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

In Proofgold the corresponding term root is 13ab2d... and proposition id is 1cefca...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__111__6

L5559

Variable x : set

(*** Conj_mul_SNo_distrR__111__6 TMKA4QnGbrP8ewmTp7yjZAd8eBK4rMndEKH bounty of about 25 bars ***)

L5560

Variable y : set

L5561

Variable z : set

L5562

Variable w : set

L5563

Variable u : set

L5564

Hypothesis H0 : SNo x

L5565

Hypothesis H1 : SNo y

L5566

Hypothesis H2 : SNo z

L5567

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5568

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5569

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5570

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5571

Hypothesis H8 : SNo (x * z)

L5572

Hypothesis H9 : SNo (y * z)

L5573

Hypothesis H10 : SNo (x * z + y * z)

L5574

Hypothesis H11 : w ∈ SNoR (x + y)

L5575

Hypothesis H12 : u ∈ SNoR z

L5576

Hypothesis H13 : SNo w

L5577

Hypothesis H14 : SNo u

L5578

Hypothesis H15 : z < u

L5579

Hypothesis H16 : SNo (x * u)

L5580

Hypothesis H17 : SNo (y * u)

L5581

Hypothesis H18 : SNo (w * z)

L5582

Hypothesis H19 : SNo ((x + y) * u)

L5583

Hypothesis H20 : SNo (w * u)

L5584

Hypothesis H21 : SNo (w * z + x * u + y * u)

L5585

Theorem. (Conj_mul_SNo_distrR__111__6)

SNo (x * z + y * z + w * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 049aa0... and proposition id is 48ba89...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__112__16

L5591

Variable x : set

(*** Conj_mul_SNo_distrR__112__16 TMTsYSd29QRVRacBeoKmMi44fv9bZkqsqt5 bounty of about 25 bars ***)

L5592

Variable y : set

L5593

Variable z : set

L5594

Variable w : set

L5595

Variable u : set

L5596

Hypothesis H0 : SNo x

L5597

Hypothesis H1 : SNo y

L5598

Hypothesis H2 : SNo z

L5599

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5600

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5601

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5602

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5603

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5604

Hypothesis H8 : SNo (x * z)

L5605

Hypothesis H9 : SNo (y * z)

L5606

Hypothesis H10 : SNo (x * z + y * z)

L5607

Hypothesis H11 : w ∈ SNoR (x + y)

L5608

Hypothesis H12 : u ∈ SNoR z

L5609

Hypothesis H13 : SNo w

L5610

Hypothesis H14 : SNo u

L5611

Hypothesis H15 : z < u

L5612

Hypothesis H17 : SNo (y * u)

L5613

Hypothesis H18 : SNo (w * z)

L5614

Hypothesis H19 : SNo ((x + y) * u)

L5615

Hypothesis H20 : SNo (w * u)

L5616

Theorem. (Conj_mul_SNo_distrR__112__16)

SNo (w * z + x * u + y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 1cb75f... and proposition id is 939e4b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__115__15

L5622

Variable x : set

(*** Conj_mul_SNo_distrR__115__15 TMdr8fpa33fDW9E7px4bSFN6qQGct1NccZR bounty of about 25 bars ***)

L5623

Variable y : set

L5624

Variable z : set

L5625

Variable w : set

L5626

Variable u : set

L5627

Hypothesis H0 : SNo x

L5628

Hypothesis H1 : SNo y

L5629

Hypothesis H2 : SNo z

L5630

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5631

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5632

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5633

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5634

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5635

Hypothesis H8 : SNo (x + y)

L5636

Hypothesis H9 : SNo (x * z)

L5637

Hypothesis H10 : SNo (y * z)

L5638

Hypothesis H11 : SNo (x * z + y * z)

L5639

Hypothesis H12 : w ∈ SNoR (x + y)

L5640

Hypothesis H13 : u ∈ SNoR z

L5641

Hypothesis H14 : SNo w

L5642

Hypothesis H16 : z < u

L5643

Hypothesis H17 : SNo (x * u)

L5644

Hypothesis H18 : SNo (y * u)

L5645

Hypothesis H19 : SNo (w * z)

L5646

Theorem. (Conj_mul_SNo_distrR__115__15)

SNo ((x + y) * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is c5df9a... and proposition id is 1462bd...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__116__7

L5652

Variable x : set

(*** Conj_mul_SNo_distrR__116__7 TMN7czdKeDMPFj3NnG9p73uF2fTNWbJtajp bounty of about 25 bars ***)

L5653

Variable y : set

L5654

Variable z : set

L5655

Variable w : set

L5656

Variable u : set

L5657

Hypothesis H0 : SNo x

L5658

Hypothesis H1 : SNo y

L5659

Hypothesis H2 : SNo z

L5660

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5661

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5662

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5663

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5664

Hypothesis H8 : SNo (x + y)

L5665

Hypothesis H9 : SNo (x * z)

L5666

Hypothesis H10 : SNo (y * z)

L5667

Hypothesis H11 : SNo (x * z + y * z)

L5668

Hypothesis H12 : w ∈ SNoR (x + y)

L5669

Hypothesis H13 : u ∈ SNoR z

L5670

Hypothesis H14 : SNo w

L5671

Hypothesis H15 : SNo u

L5672

Hypothesis H16 : z < u

L5673

Hypothesis H17 : SNo (x * u)

L5674

Hypothesis H18 : SNo (y * u)

L5675

Theorem. (Conj_mul_SNo_distrR__116__7)

SNo (w * z) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 77401f... and proposition id is 1918ff...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__116__15

L5681

Variable x : set

(*** Conj_mul_SNo_distrR__116__15 TMKyhHdRiSpxLyfSq3VUGSMRfSwctKQycDi bounty of about 25 bars ***)

L5682

Variable y : set

L5683

Variable z : set

L5684

Variable w : set

L5685

Variable u : set

L5686

Hypothesis H0 : SNo x

L5687

Hypothesis H1 : SNo y

L5688

Hypothesis H2 : SNo z

L5689

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5690

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5691

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5692

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5693

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5694

Hypothesis H8 : SNo (x + y)

L5695

Hypothesis H9 : SNo (x * z)

L5696

Hypothesis H10 : SNo (y * z)

L5697

Hypothesis H11 : SNo (x * z + y * z)

L5698

Hypothesis H12 : w ∈ SNoR (x + y)

L5699

Hypothesis H13 : u ∈ SNoR z

L5700

Hypothesis H14 : SNo w

L5701

Hypothesis H16 : z < u

L5702

Hypothesis H17 : SNo (x * u)

L5703

Hypothesis H18 : SNo (y * u)

L5704

Theorem. (Conj_mul_SNo_distrR__116__15)

SNo (w * z) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 936f0f... and proposition id is f33b12...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__117__12

L5710

Variable x : set

(*** Conj_mul_SNo_distrR__117__12 TMUEPZtJ8VRUMpL7GBoiZDjMtTJZq1k5edf bounty of about 25 bars ***)

L5711

Variable y : set

L5712

Variable z : set

L5713

Variable w : set

L5714

Variable u : set

L5715

Hypothesis H0 : SNo x

L5716

Hypothesis H1 : SNo y

L5717

Hypothesis H2 : SNo z

L5718

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5719

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5720

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5721

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5722

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5723

Hypothesis H8 : SNo (x + y)

L5724

Hypothesis H9 : SNo (x * z)

L5725

Hypothesis H10 : SNo (y * z)

L5726

Hypothesis H11 : SNo (x * z + y * z)

L5727

Hypothesis H13 : u ∈ SNoR z

L5728

Hypothesis H14 : SNo w

L5729

Hypothesis H15 : SNo u

L5730

Hypothesis H16 : z < u

L5731

Hypothesis H17 : SNo (x * u)

L5732

Theorem. (Conj_mul_SNo_distrR__117__12)

SNo (y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 51c283... and proposition id is b33961...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__118__12

L5738

Variable x : set

(*** Conj_mul_SNo_distrR__118__12 TMErf8zaukDoYseiSjUgjpyqPn9NCEmjXGK bounty of about 25 bars ***)

L5739

Variable y : set

L5740

Variable z : set

L5741

Variable w : set

L5742

Variable u : set

L5743

Hypothesis H0 : SNo x

L5744

Hypothesis H1 : SNo y

L5745

Hypothesis H2 : SNo z

L5746

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5747

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5748

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5749

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5750

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5751

Hypothesis H8 : SNo (x + y)

L5752

Hypothesis H9 : SNo (x * z)

L5753

Hypothesis H10 : SNo (y * z)

L5754

Hypothesis H11 : SNo (x * z + y * z)

L5755

Hypothesis H13 : u ∈ SNoR z

L5756

Hypothesis H14 : SNo w

L5757

Hypothesis H15 : SNo u

L5758

Hypothesis H16 : z < u

L5759

Theorem. (Conj_mul_SNo_distrR__118__12)

SNo (x * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is a76d02... and proposition id is 166915...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__119__9

L5765

Variable x : set

(*** Conj_mul_SNo_distrR__119__9 TMVSquwN6wmCxEKjoGQgbEGZgucA73LggWW bounty of about 25 bars ***)

L5766

Variable y : set

L5767

Variable z : set

L5768

Variable w : set

L5769

Variable u : set

L5770

Variable v : set

L5771

Hypothesis H0 : SNo x

L5772

Hypothesis H1 : SNo y

L5773

Hypothesis H2 : SNo z

L5774

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L5775

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L5776

Hypothesis H5 : SNo (x * z)

L5777

Hypothesis H6 : SNo (y * z)

L5778

Hypothesis H7 : u ∈ SNoL z

L5779

Hypothesis H8 : SNo w

L5780

Hypothesis H10 : u < z

L5781

Hypothesis H11 : SNo (x * u)

L5782

Hypothesis H12 : SNo (y * u)

L5783

Hypothesis H13 : SNo (w * z)

L5784

Hypothesis H14 : SNo (w * u)

L5785

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5786

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5787

Hypothesis H17 : v ∈ SNoL y

L5788

Hypothesis H18 : w ≤ x + v

L5789

Hypothesis H19 : SNo v

L5790

Hypothesis H20 : v < y

L5791

Hypothesis H21 : SNo (v * u)

L5792

Hypothesis H22 : SNo (v * z)

L5793

Theorem. (Conj_mul_SNo_distrR__119__9)

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

In Proofgold the corresponding term root is 1b6fec... and proposition id is 579dbb...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__120__16

L5799

Variable x : set

(*** Conj_mul_SNo_distrR__120__16 TMSuPWeJog6BvBemPtjpDyM3kJsRxpDJrpo bounty of about 25 bars ***)

L5800

Variable y : set

L5801

Variable z : set

L5802

Variable w : set

L5803

Variable u : set

L5804

Variable v : set

L5805

Hypothesis H0 : SNo x

L5806

Hypothesis H1 : SNo y

L5807

Hypothesis H2 : SNo z

L5808

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (x + x2) * z = x * z + x2 * z)

L5809

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L5810

Hypothesis H5 : SNo (x * z)

L5811

Hypothesis H6 : SNo (y * z)

L5812

Hypothesis H7 : u ∈ SNoL z

L5813

Hypothesis H8 : SNo w

L5814

Hypothesis H9 : SNo u

L5815

Hypothesis H10 : u < z

L5816

Hypothesis H11 : SNo (x * u)

L5817

Hypothesis H12 : SNo (y * u)

L5818

Hypothesis H13 : SNo (w * z)

L5819

Hypothesis H14 : SNo (w * u)

L5820

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5821

Hypothesis H17 : v ∈ SNoL y

L5822

Hypothesis H18 : w ≤ x + v

L5823

Hypothesis H19 : SNo v

L5824

Hypothesis H20 : v < y

L5825

Hypothesis H21 : SNo (v * u)

L5826

Theorem. (Conj_mul_SNo_distrR__120__16)

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

In Proofgold the corresponding term root is ab6df9... and proposition id is edda61...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__121__3

L5832

Variable x : set

(*** Conj_mul_SNo_distrR__121__3 TMVzGEdKhSYAVyKfTJN9ubZcnnifuK6XCxX bounty of about 25 bars ***)

L5833

Variable y : set

L5834

Variable z : set

L5835

Variable w : set

L5836

Variable u : set

L5837

Variable v : set

L5838

Hypothesis H0 : SNo x

L5839

Hypothesis H1 : SNo y

L5840

Hypothesis H2 : SNo z

L5841

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x + x2) * y2 = x * y2 + x2 * y2))

L5842

Hypothesis H5 : SNo (x * z)

L5843

Hypothesis H6 : SNo (y * z)

L5844

Hypothesis H7 : u ∈ SNoL z

L5845

Hypothesis H8 : SNo w

L5846

Hypothesis H9 : SNo u

L5847

Hypothesis H10 : u < z

L5848

Hypothesis H11 : SNo (x * u)

L5849

Hypothesis H12 : SNo (y * u)

L5850

Hypothesis H13 : SNo (w * z)

L5851

Hypothesis H14 : SNo (w * u)

L5852

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5853

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5854

Hypothesis H17 : v ∈ SNoL y

L5855

Hypothesis H18 : w ≤ x + v

L5856

Hypothesis H19 : SNo v

L5857

Hypothesis H20 : v < y

L5858

Theorem. (Conj_mul_SNo_distrR__121__3)

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

In Proofgold the corresponding term root is d8ed5f... and proposition id is 81db66...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__123__10

L5864

Variable x : set

(*** Conj_mul_SNo_distrR__123__10 TMHb1ZGudXqMwXcWmR5PPAcJwWxTt7GX8K6 bounty of about 25 bars ***)

L5865

Variable y : set

L5866

Variable z : set

L5867

Variable w : set

L5868

Variable u : set

L5869

Variable v : set

L5870

Hypothesis H0 : SNo x

L5871

Hypothesis H1 : SNo y

L5872

Hypothesis H2 : SNo z

L5873

Hypothesis H3 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (x2 + y) * z = x2 * z + y * z)

L5874

Hypothesis H4 : (∀x2 : set, x2 ∈ SNoS_ (SNoLev x) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x2 + y) * y2 = x2 * y2 + y * y2))

L5875

Hypothesis H5 : SNo (x * z)

L5876

Hypothesis H6 : SNo (y * z)

L5877

Hypothesis H7 : u ∈ SNoL z

L5878

Hypothesis H8 : SNo w

L5879

Hypothesis H9 : SNo u

L5880

Hypothesis H11 : SNo (x * u)

L5881

Hypothesis H12 : SNo (y * u)

L5882

Hypothesis H13 : SNo (w * z)

L5883

Hypothesis H14 : SNo (w * u)

L5884

Hypothesis H15 : SNo (w * z + x * u + y * u)

L5885

Hypothesis H16 : SNo (x * z + y * z + w * u)

L5886

Hypothesis H17 : v ∈ SNoL x

L5887

Hypothesis H18 : w ≤ v + y

L5888

Hypothesis H19 : SNo v

L5889

Hypothesis H20 : v < x

L5890

Hypothesis H21 : SNo (v * u)

L5891

Theorem. (Conj_mul_SNo_distrR__123__10)

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

In Proofgold the corresponding term root is 484f15... and proposition id is f94847...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__125__15

L5897

Variable x : set

(*** Conj_mul_SNo_distrR__125__15 TMG6Ed7CNp5CtxzisUGSLKRzaAg8jb8PHsA bounty of about 25 bars ***)

L5898

Variable y : set

L5899

Variable z : set

L5900

Variable w : set

L5901

Variable u : set

L5902

Hypothesis H0 : SNo x

L5903

Hypothesis H1 : SNo y

L5904

Hypothesis H2 : SNo z

L5905

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5906

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5907

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5908

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5909

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5910

Hypothesis H8 : SNo (x * z)

L5911

Hypothesis H9 : SNo (y * z)

L5912

Hypothesis H10 : SNo (x * z + y * z)

L5913

Hypothesis H11 : w ∈ SNoL (x + y)

L5914

Hypothesis H12 : u ∈ SNoL z

L5915

Hypothesis H13 : SNo w

L5916

Hypothesis H14 : SNo u

L5917

Hypothesis H16 : SNo (x * u)

L5918

Hypothesis H17 : SNo (y * u)

L5919

Hypothesis H18 : SNo (w * z)

L5920

Hypothesis H19 : SNo ((x + y) * u)

L5921

Hypothesis H20 : SNo (w * u)

L5922

Hypothesis H21 : SNo (w * z + x * u + y * u)

L5923

Theorem. (Conj_mul_SNo_distrR__125__15)

SNo (x * z + y * z + w * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 74b62e... and proposition id is 280c71...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__126__9

L5929

Variable x : set

(*** Conj_mul_SNo_distrR__126__9 TMS9ZBnEU5zJXypDnr55dyZj2vQoNMPXJ12 bounty of about 25 bars ***)

L5930

Variable y : set

L5931

Variable z : set

L5932

Variable w : set

L5933

Variable u : set

L5934

Hypothesis H0 : SNo x

L5935

Hypothesis H1 : SNo y

L5936

Hypothesis H2 : SNo z

L5937

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5938

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5939

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5940

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5941

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5942

Hypothesis H8 : SNo (x * z)

L5943

Hypothesis H10 : SNo (x * z + y * z)

L5944

Hypothesis H11 : w ∈ SNoL (x + y)

L5945

Hypothesis H12 : u ∈ SNoL z

L5946

Hypothesis H13 : SNo w

L5947

Hypothesis H14 : SNo u

L5948

Hypothesis H15 : u < z

L5949

Hypothesis H16 : SNo (x * u)

L5950

Hypothesis H17 : SNo (y * u)

L5951

Hypothesis H18 : SNo (w * z)

L5952

Hypothesis H19 : SNo ((x + y) * u)

L5953

Hypothesis H20 : SNo (w * u)

L5954

Theorem. (Conj_mul_SNo_distrR__126__9)

SNo (w * z + x * u + y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 6b83fb... and proposition id is a2b2d0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__126__11

L5960

Variable x : set

(*** Conj_mul_SNo_distrR__126__11 TMQQ5TytRWd5jQx4kwLCNYzJX3XtGoNx838 bounty of about 25 bars ***)

L5961

Variable y : set

L5962

Variable z : set

L5963

Variable w : set

L5964

Variable u : set

L5965

Hypothesis H0 : SNo x

L5966

Hypothesis H1 : SNo y

L5967

Hypothesis H2 : SNo z

L5968

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L5969

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L5970

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L5971

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L5972

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L5973

Hypothesis H8 : SNo (x * z)

L5974

Hypothesis H9 : SNo (y * z)

L5975

Hypothesis H10 : SNo (x * z + y * z)

L5976

Hypothesis H12 : u ∈ SNoL z

L5977

Hypothesis H13 : SNo w

L5978

Hypothesis H14 : SNo u

L5979

Hypothesis H15 : u < z

L5980

Hypothesis H16 : SNo (x * u)

L5981

Hypothesis H17 : SNo (y * u)

L5982

Hypothesis H18 : SNo (w * z)

L5983

Hypothesis H19 : SNo ((x + y) * u)

L5984

Hypothesis H20 : SNo (w * u)

L5985

Theorem. (Conj_mul_SNo_distrR__126__11)

SNo (w * z + x * u + y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is ab7ed6... and proposition id is 4590ea...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__127__7

L5991

Variable x : set

(*** Conj_mul_SNo_distrR__127__7 TMFRFLNCAwByM8QsRVEQ8hBkoSgiMXU6uRX bounty of about 25 bars ***)

L5992

Variable y : set

L5993

Variable z : set

L5994

Variable w : set

L5995

Variable u : set

L5996

Hypothesis H0 : SNo x

L5997

Hypothesis H1 : SNo y

L5998

Hypothesis H2 : SNo z

L5999

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6000

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6001

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6002

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6003

Hypothesis H8 : SNo (x * z)

L6004

Hypothesis H9 : SNo (y * z)

L6005

Hypothesis H10 : SNo (x * z + y * z)

L6006

Hypothesis H11 : w ∈ SNoL (x + y)

L6007

Hypothesis H12 : u ∈ SNoL z

L6008

Hypothesis H13 : SNo w

L6009

Hypothesis H14 : SNo u

L6010

Hypothesis H15 : u < z

L6011

Hypothesis H16 : SNo (x * u)

L6012

Hypothesis H17 : SNo (y * u)

L6013

Hypothesis H18 : SNo (w * z)

L6014

Hypothesis H19 : SNo ((x + y) * u)

L6015

Theorem. (Conj_mul_SNo_distrR__127__7)

SNo (w * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is e1715c... and proposition id is d72eca...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__128__15

L6021

Variable x : set

(*** Conj_mul_SNo_distrR__128__15 TMdio434wQ2X1DSMfMBLroMwX78BTLPX4xb bounty of about 25 bars ***)

L6022

Variable y : set

L6023

Variable z : set

L6024

Variable w : set

L6025

Variable u : set

L6026

Hypothesis H0 : SNo x

L6027

Hypothesis H1 : SNo y

L6028

Hypothesis H2 : SNo z

L6029

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6030

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6031

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6032

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6033

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6034

Hypothesis H8 : SNo (x * z)

L6035

Hypothesis H9 : SNo (y * z)

L6036

Hypothesis H10 : SNo (x * z + y * z)

L6037

Hypothesis H11 : w ∈ SNoL (x + y)

L6038

Hypothesis H12 : u ∈ SNoL z

L6039

Hypothesis H13 : SNo w

L6040

Hypothesis H14 : SNo u

L6041

Hypothesis H16 : SNo (x * u)

L6042

Hypothesis H17 : SNo (y * u)

L6043

Hypothesis H18 : SNo (w * z)

L6044

Hypothesis H19 : SNo ((x + y) * u)

L6045

Theorem. (Conj_mul_SNo_distrR__128__15)

SNo (x * u + y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 1ede90... and proposition id is a537df...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__129__10

L6051

Variable x : set

(*** Conj_mul_SNo_distrR__129__10 TMbCVvbWdvQuvVDWx1pmoiYFB9xdSpfTUAu bounty of about 25 bars ***)

L6052

Variable y : set

L6053

Variable z : set

L6054

Variable w : set

L6055

Variable u : set

L6056

Hypothesis H0 : SNo x

L6057

Hypothesis H1 : SNo y

L6058

Hypothesis H2 : SNo z

L6059

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6060

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6061

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6062

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6063

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6064

Hypothesis H8 : SNo (x + y)

L6065

Hypothesis H9 : SNo (x * z)

L6066

Hypothesis H11 : SNo (x * z + y * z)

L6067

Hypothesis H12 : w ∈ SNoL (x + y)

L6068

Hypothesis H13 : u ∈ SNoL z

L6069

Hypothesis H14 : SNo w

L6070

Hypothesis H15 : SNo u

L6071

Hypothesis H16 : u < z

L6072

Hypothesis H17 : SNo (x * u)

L6073

Hypothesis H18 : SNo (y * u)

L6074

Hypothesis H19 : SNo (w * z)

L6075

Theorem. (Conj_mul_SNo_distrR__129__10)

SNo ((x + y) * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 1d2393... and proposition id is dd4011...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__130__13

L6081

Variable x : set

(*** Conj_mul_SNo_distrR__130__13 TMFqsC1YdjeodiemfS8aTPvwvqXBma6DZjj bounty of about 25 bars ***)

L6082

Variable y : set

L6083

Variable z : set

L6084

Variable w : set

L6085

Variable u : set

L6086

Hypothesis H0 : SNo x

L6087

Hypothesis H1 : SNo y

L6088

Hypothesis H2 : SNo z

L6089

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6090

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6091

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6092

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6093

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6094

Hypothesis H8 : SNo (x + y)

L6095

Hypothesis H9 : SNo (x * z)

L6096

Hypothesis H10 : SNo (y * z)

L6097

Hypothesis H11 : SNo (x * z + y * z)

L6098

Hypothesis H12 : w ∈ SNoL (x + y)

L6099

Hypothesis H14 : SNo w

L6100

Hypothesis H15 : SNo u

L6101

Hypothesis H16 : u < z

L6102

Hypothesis H17 : SNo (x * u)

L6103

Hypothesis H18 : SNo (y * u)

L6104

Theorem. (Conj_mul_SNo_distrR__130__13)

SNo (w * z) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 7cb760... and proposition id is 86c358...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__131__12

L6110

Variable x : set

(*** Conj_mul_SNo_distrR__131__12 TMbk48qRYLWf1AtxikXuWdBGhZnFdyeikbT bounty of about 25 bars ***)

L6111

Variable y : set

L6112

Variable z : set

L6113

Variable w : set

L6114

Variable u : set

L6115

Hypothesis H0 : SNo x

L6116

Hypothesis H1 : SNo y

L6117

Hypothesis H2 : SNo z

L6118

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6119

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6120

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6121

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6122

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6123

Hypothesis H8 : SNo (x + y)

L6124

Hypothesis H9 : SNo (x * z)

L6125

Hypothesis H10 : SNo (y * z)

L6126

Hypothesis H11 : SNo (x * z + y * z)

L6127

Hypothesis H13 : u ∈ SNoL z

L6128

Hypothesis H14 : SNo w

L6129

Hypothesis H15 : SNo u

L6130

Hypothesis H16 : u < z

L6131

Hypothesis H17 : SNo (x * u)

L6132

Theorem. (Conj_mul_SNo_distrR__131__12)

SNo (y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 351a8e... and proposition id is 9d2c72...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__131__17

L6138

Variable x : set

(*** Conj_mul_SNo_distrR__131__17 TMXVxFvrAEo8u5qotzvuZkF3E1zE7NX9eqh bounty of about 25 bars ***)

L6139

Variable y : set

L6140

Variable z : set

L6141

Variable w : set

L6142

Variable u : set

L6143

Hypothesis H0 : SNo x

L6144

Hypothesis H1 : SNo y

L6145

Hypothesis H2 : SNo z

L6146

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6147

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6148

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6149

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6150

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6151

Hypothesis H8 : SNo (x + y)

L6152

Hypothesis H9 : SNo (x * z)

L6153

Hypothesis H10 : SNo (y * z)

L6154

Hypothesis H11 : SNo (x * z + y * z)

L6155

Hypothesis H12 : w ∈ SNoL (x + y)

L6156

Hypothesis H13 : u ∈ SNoL z

L6157

Hypothesis H14 : SNo w

L6158

Hypothesis H15 : SNo u

L6159

Hypothesis H16 : u < z

L6160

Theorem. (Conj_mul_SNo_distrR__131__17)

SNo (y * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is f6b991... and proposition id is 74b966...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__132__4

L6166

Variable x : set

(*** Conj_mul_SNo_distrR__132__4 TMYoVokjyoKoJZZoXWnadx6zKwz8jh3B98L bounty of about 25 bars ***)

L6167

Variable y : set

L6168

Variable z : set

L6169

Variable w : set

L6170

Variable u : set

L6171

Hypothesis H0 : SNo x

L6172

Hypothesis H1 : SNo y

L6173

Hypothesis H2 : SNo z

L6174

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6175

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6176

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6177

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6178

Hypothesis H8 : SNo (x + y)

L6179

Hypothesis H9 : SNo (x * z)

L6180

Hypothesis H10 : SNo (y * z)

L6181

Hypothesis H11 : SNo (x * z + y * z)

L6182

Hypothesis H12 : w ∈ SNoL (x + y)

L6183

Hypothesis H13 : u ∈ SNoL z

L6184

Hypothesis H14 : SNo w

L6185

Hypothesis H15 : SNo u

L6186

Hypothesis H16 : u < z

L6187

Theorem. (Conj_mul_SNo_distrR__132__4)

SNo (x * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 76a4d6... and proposition id is 8cf704...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__132__16

L6193

Variable x : set

(*** Conj_mul_SNo_distrR__132__16 TMV6XTChegLQBsFFGLeXpn9qjfcsuYrvDJA bounty of about 25 bars ***)

L6194

Variable y : set

L6195

Variable z : set

L6196

Variable w : set

L6197

Variable u : set

L6198

Hypothesis H0 : SNo x

L6199

Hypothesis H1 : SNo y

L6200

Hypothesis H2 : SNo z

L6201

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6202

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6203

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6204

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6205

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6206

Hypothesis H8 : SNo (x + y)

L6207

Hypothesis H9 : SNo (x * z)

L6208

Hypothesis H10 : SNo (y * z)

L6209

Hypothesis H11 : SNo (x * z + y * z)

L6210

Hypothesis H12 : w ∈ SNoL (x + y)

L6211

Hypothesis H13 : u ∈ SNoL z

L6212

Hypothesis H14 : SNo w

L6213

Hypothesis H15 : SNo u

L6214

Theorem. (Conj_mul_SNo_distrR__132__16)

SNo (x * u) → (w * z + (x + y) * u + - (w * u)) < x * z + y * z

In Proofgold the corresponding term root is 055a33... and proposition id is 65b864...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__133__11

L6220

Variable x : set

(*** Conj_mul_SNo_distrR__133__11 TML7HZcZ7JwBWG2K9ezf3QaNvpogEzhkBTX bounty of about 25 bars ***)

L6221

Variable y : set

L6222

Variable z : set

L6223

Variable w : set

L6224

Variable u : set

L6225

Hypothesis H0 : SNo x

L6226

Hypothesis H1 : SNo y

L6227

Hypothesis H2 : SNo z

L6228

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6229

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6230

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6231

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6232

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6233

Hypothesis H8 : SNoCutP w u

L6234

Hypothesis H9 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL (x + y) → (∀y2 : set, y2 ∈ SNoL z → v = x2 * z + (x + y) * y2 + - (x2 * y2) → P)) → (∀x2 : set, x2 ∈ SNoR (x + y) → (∀y2 : set, y2 ∈ SNoR z → v = x2 * z + (x + y) * y2 + - (x2 * y2) → P)) → P))

L6235

Hypothesis H10 : (∀v : set, v ∈ u → (∀P : prop, (∀x2 : set, x2 ∈ SNoL (x + y) → (∀y2 : set, y2 ∈ SNoR z → v = x2 * z + (x + y) * y2 + - (x2 * y2) → P)) → (∀x2 : set, x2 ∈ SNoR (x + y) → (∀y2 : set, y2 ∈ SNoL z → v = x2 * z + (x + y) * y2 + - (x2 * y2) → P)) → P))

L6236

Hypothesis H12 : SNo (x + y)

L6237

Hypothesis H13 : SNo ((x + y) * z)

L6238

Hypothesis H14 : SNo (x * z)

L6239

Hypothesis H15 : SNo (y * z)

L6240

Hypothesis H16 : SNo (x * z + y * z)

L6241

Theorem. (Conj_mul_SNo_distrR__133__11)

x * z + y * z = SNoCut (binunion (Repl (SNoL (x * z)) (λv : set ⇒ v + y * z)) (Repl (SNoL (y * z)) (add_SNo (x * z)))) (binunion (Repl (SNoR (x * z)) (λv : set ⇒ v + y * z)) (Repl (SNoR (y * z)) (add_SNo (x * z)))) → (x + y) * z = x * z + y * z

In Proofgold the corresponding term root is 8270ad... and proposition id is 0de53b...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__135__6

L6247

Variable x : set

(*** Conj_mul_SNo_distrR__135__6 TMUdLTNFkPPLEBZomZJEP1p5q1yPFnJUrCD bounty of about 25 bars ***)

L6248

Variable y : set

L6249

Variable z : set

L6250

Variable w : set

L6251

Variable u : set

L6252

Hypothesis H0 : SNo x

L6253

Hypothesis H1 : SNo y

L6254

Hypothesis H2 : SNo z

L6255

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6256

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6257

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6258

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6259

Hypothesis H8 : SNoCutP w u

L6260

L6261

L6262

Hypothesis H11 : (x + y) * z = SNoCut w u

L6263

Hypothesis H12 : SNo (x + y)

L6264

Hypothesis H13 : SNo ((x + y) * z)

L6265

Hypothesis H14 : SNo (x * z)

L6266

Theorem. (Conj_mul_SNo_distrR__135__6)

SNo (y * z) → (x + y) * z = x * z + y * z

In Proofgold the corresponding term root is df6690... and proposition id is 525cac...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__136__5

L6272

Variable x : set

(*** Conj_mul_SNo_distrR__136__5 TMSnYvLozqEjEtvJrXtGzfQjDkyd1f5Vs7W bounty of about 25 bars ***)

L6273

Variable y : set

L6274

Variable z : set

L6275

Variable w : set

L6276

Variable u : set

L6277

Hypothesis H0 : SNo x

L6278

Hypothesis H1 : SNo y

L6279

Hypothesis H2 : SNo z

L6280

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6281

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6282

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6283

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6284

Hypothesis H8 : SNoCutP w u

L6285

L6286

L6287

Hypothesis H11 : (x + y) * z = SNoCut w u

L6288

Hypothesis H12 : SNo (x + y)

L6289

Hypothesis H13 : SNo ((x + y) * z)

L6290

Theorem. (Conj_mul_SNo_distrR__136__5)

SNo (x * z) → (x + y) * z = x * z + y * z

In Proofgold the corresponding term root is 32c758... and proposition id is e37909...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__137__3

L6296

Variable x : set

(*** Conj_mul_SNo_distrR__137__3 TMczyWUXwud5rYRj376zaqvmmeCKZRMfCj4 bounty of about 25 bars ***)

L6297

Variable y : set

L6298

Variable z : set

L6299

Variable w : set

L6300

Variable u : set

L6301

Hypothesis H0 : SNo x

L6302

Hypothesis H1 : SNo y

L6303

Hypothesis H2 : SNo z

L6304

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6305

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6306

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6307

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6308

Hypothesis H8 : SNoCutP w u

L6309

L6310

L6311

Hypothesis H11 : (x + y) * z = SNoCut w u

L6312

Hypothesis H12 : SNo (x + y)

L6313

Theorem. (Conj_mul_SNo_distrR__137__3)

SNo ((x + y) * z) → (x + y) * z = x * z + y * z

In Proofgold the corresponding term root is bbe03c... and proposition id is 80b487...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__137__4

L6319

Variable x : set

(*** Conj_mul_SNo_distrR__137__4 TMdgaUgt6mi2ReHtWvwJifq4ETmGwGbiY7b bounty of about 25 bars ***)

L6320

Variable y : set

L6321

Variable z : set

L6322

Variable w : set

L6323

Variable u : set

L6324

Hypothesis H0 : SNo x

L6325

Hypothesis H1 : SNo y

L6326

Hypothesis H2 : SNo z

L6327

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6328

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6329

Hypothesis H6 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (v + y) * x2 = v * x2 + y * x2))

L6330

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6331

Hypothesis H8 : SNoCutP w u

L6332

L6333

L6334

Hypothesis H11 : (x + y) * z = SNoCut w u

L6335

Hypothesis H12 : SNo (x + y)

L6336

Theorem. (Conj_mul_SNo_distrR__137__4)

SNo ((x + y) * z) → (x + y) * z = x * z + y * z

In Proofgold the corresponding term root is cd3e5a... and proposition id is b6359c...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_distrR__138__6

L6342

Variable x : set

(*** Conj_mul_SNo_distrR__138__6 TMQdquZdAwnr9iL7hp3Cv2QXzBuKjSyKXRN bounty of about 25 bars ***)

L6343

Variable y : set

L6344

Variable z : set

L6345

Variable w : set

L6346

Variable u : set

L6347

Hypothesis H0 : SNo x

L6348

Hypothesis H1 : SNo y

L6349

Hypothesis H2 : SNo z

L6350

Hypothesis H3 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (v + y) * z = v * z + y * z)

L6351

Hypothesis H4 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (x + v) * z = x * z + v * z)

L6352

Hypothesis H5 : (∀v : set, v ∈ SNoS_ (SNoLev z) → (x + y) * v = x * v + y * v)

L6353

Hypothesis H7 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → (x + v) * x2 = x * x2 + v * x2))

L6354

Hypothesis H8 : SNoCutP w u

L6355

L6356

L6357

Hypothesis H11 : (x + y) * z = SNoCut w u

L6358

Theorem. (Conj_mul_SNo_distrR__138__6)

SNo (x + y) → (x + y) * z = x * z + y * z

In Proofgold the corresponding term root is 6aef7c... and proposition id is b01fc4...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__1__7

L6364

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__1__7 TMKnnoPW6hcBe3TD7vdUN6JmCK4pKEW7jMc bounty of about 25 bars ***)

L6365

Variable x : set

L6366

Variable y : set

L6367

Variable z : set

L6368

Variable w : set

L6369

Variable u : set

L6370

Variable v : set

L6371

Variable x2 : set

L6372

Variable y2 : set

L6373

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L6374

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

L6375

Hypothesis H2 : SNo x

L6376

Hypothesis H3 : SNo z

L6377

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6378

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6379

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6380

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6381

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6382

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)))

L6383

Hypothesis H11 : SNo (g x y)

L6384

Hypothesis H12 : SNo (g (g x y) z)

L6385

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L6386

Hypothesis H14 : SNo v

L6387

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L6388

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L6389

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L6390

Hypothesis H18 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6391

Hypothesis H19 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6392

Hypothesis H20 : SNo u

L6393

Hypothesis H21 : SNo y2

L6394

Hypothesis H22 : SNo (g u (g y z))

L6395

Hypothesis H23 : SNo (g x v)

L6396

Hypothesis H24 : SNo (g x x2)

L6397

Hypothesis H25 : SNo (g u v)

L6398

Hypothesis H26 : SNo (g u x2)

L6399

Hypothesis H27 : SNo (g u y)

L6400

Hypothesis H28 : SNo (g x2 z)

L6401

Hypothesis H29 : SNo (g y y2)

L6402

Hypothesis H30 : SNo (g u (g x2 z))

L6403

Hypothesis H31 : SNo (g u (g y y2))

L6404

Hypothesis H32 : SNo (g x2 y2)

L6405

Hypothesis H33 : SNo (g x (g x2 y2))

L6406

Hypothesis H34 : SNo (g x (g x2 z))

L6407

Hypothesis H35 : SNo (g x (g y y2))

L6408

Hypothesis H36 : SNo (g u (g y z) + g x v)

L6409

Hypothesis H37 : SNo (g (g x y) z + g u v)

L6410

Hypothesis H38 : SNo (g u (g x2 y2))

L6411

Hypothesis H39 : SNo (g u (v + g x2 y2))

L6412

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L6413

Hypothesis H41 : SNo (g u (g y z) + g x (g x2 z) + g x (g y y2) + g u v + g u (g x2 y2))

L6414

Theorem. (Conj_mul_SNo_assoc_lem1__1__7)

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

In Proofgold the corresponding term root is a5ff4a... and proposition id is 9c76fa...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__1__33

L6420

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__1__33 TMZ2dwQaMb41nvs4kPygdgDzVBerbqzWcPV bounty of about 25 bars ***)

L6421

Variable x : set

L6422

Variable y : set

L6423

Variable z : set

L6424

Variable w : set

L6425

Variable u : set

L6426

Variable v : set

L6427

Variable x2 : set

L6428

Variable y2 : set

L6429

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L6430

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

L6431

Hypothesis H2 : SNo x

L6432

Hypothesis H3 : SNo z

L6433

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6434

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6435

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6436

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6437

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6438

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6439

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)))

L6440

Hypothesis H11 : SNo (g x y)

L6441

Hypothesis H12 : SNo (g (g x y) z)

L6442

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L6443

Hypothesis H14 : SNo v

L6444

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L6445

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L6446

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L6447

Hypothesis H18 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6448

Hypothesis H19 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6449

Hypothesis H20 : SNo u

L6450

Hypothesis H21 : SNo y2

L6451

Hypothesis H22 : SNo (g u (g y z))

L6452

Hypothesis H23 : SNo (g x v)

L6453

Hypothesis H24 : SNo (g x x2)

L6454

Hypothesis H25 : SNo (g u v)

L6455

Hypothesis H26 : SNo (g u x2)

L6456

Hypothesis H27 : SNo (g u y)

L6457

Hypothesis H28 : SNo (g x2 z)

L6458

Hypothesis H29 : SNo (g y y2)

L6459

Hypothesis H30 : SNo (g u (g x2 z))

L6460

Hypothesis H31 : SNo (g u (g y y2))

L6461

Hypothesis H32 : SNo (g x2 y2)

L6462

Hypothesis H34 : SNo (g x (g x2 z))

L6463

Hypothesis H35 : SNo (g x (g y y2))

L6464

Hypothesis H36 : SNo (g u (g y z) + g x v)

L6465

Hypothesis H37 : SNo (g (g x y) z + g u v)

L6466

Hypothesis H38 : SNo (g u (g x2 y2))

L6467

Hypothesis H39 : SNo (g u (v + g x2 y2))

L6468

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L6469

Hypothesis H41 : SNo (g u (g y z) + g x (g x2 z) + g x (g y y2) + g u v + g u (g x2 y2))

L6470

Theorem. (Conj_mul_SNo_assoc_lem1__1__33)

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

In Proofgold the corresponding term root is 3c6998... and proposition id is 6f642f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__2__27

L6476

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__2__27 TMMYbjBBEyXjofdffuhoRfHHJBgAska7jJo bounty of about 25 bars ***)

L6477

Variable x : set

L6478

Variable y : set

L6479

Variable z : set

L6480

Variable w : set

L6481

Variable u : set

L6482

Variable v : set

L6483

Variable x2 : set

L6484

Variable y2 : set

L6485

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L6486

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

L6487

Hypothesis H2 : SNo x

L6488

Hypothesis H3 : SNo z

L6489

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6490

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6491

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6492

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6493

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6494

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6495

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)))

L6496

Hypothesis H11 : SNo (g x y)

L6497

Hypothesis H12 : SNo (g (g x y) z)

L6498

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L6499

Hypothesis H14 : SNo v

L6500

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L6501

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L6502

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L6503

Hypothesis H18 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6504

Hypothesis H19 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6505

Hypothesis H20 : SNo u

L6506

Hypothesis H21 : SNo y2

L6507

Hypothesis H22 : SNo (g u (g y z))

L6508

Hypothesis H23 : SNo (g x v)

L6509

Hypothesis H24 : SNo (g x x2)

L6510

Hypothesis H25 : SNo (g u v)

L6511

Hypothesis H26 : SNo (g u x2)

L6512

Hypothesis H28 : SNo (g x2 z)

L6513

Hypothesis H29 : SNo (g y y2)

L6514

Hypothesis H30 : SNo (g u (g x2 z))

L6515

Hypothesis H31 : SNo (g u (g y y2))

L6516

Hypothesis H32 : SNo (g x2 y2)

L6517

Hypothesis H33 : SNo (g x (g x2 y2))

L6518

Hypothesis H34 : SNo (g x (g x2 z))

L6519

Hypothesis H35 : SNo (g x (g y y2))

L6520

Hypothesis H36 : SNo (g u (g y z) + g x v)

L6521

Hypothesis H37 : SNo (g (g x y) z + g u v)

L6522

Hypothesis H38 : SNo (g u (g x2 y2))

L6523

Hypothesis H39 : SNo (g u (v + g x2 y2))

L6524

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L6525

Hypothesis H41 : SNo (g u v + g u (g x2 y2))

L6526

Theorem. (Conj_mul_SNo_assoc_lem1__2__27)

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

In Proofgold the corresponding term root is 8d1808... and proposition id is ff4477...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__3__17

L6532

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__3__17 TMN4vHLkZcrXP7tS5GZpmPUBoN4YsEqLyv5 bounty of about 25 bars ***)

L6533

Variable x : set

L6534

Variable y : set

L6535

Variable z : set

L6536

Variable w : set

L6537

Variable u : set

L6538

Variable v : set

L6539

Variable x2 : set

L6540

Variable y2 : set

L6541

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L6542

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

L6543

Hypothesis H2 : SNo x

L6544

Hypothesis H3 : SNo z

L6545

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6546

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6547

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6548

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6549

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6550

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6551

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)))

L6552

Hypothesis H11 : SNo (g x y)

L6553

Hypothesis H12 : SNo (g (g x y) z)

L6554

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L6555

Hypothesis H14 : SNo v

L6556

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L6557

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L6558

Hypothesis H18 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6559

Hypothesis H19 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6560

Hypothesis H20 : SNo u

L6561

Hypothesis H21 : SNo y2

L6562

Hypothesis H22 : SNo (g u (g y z))

L6563

Hypothesis H23 : SNo (g x v)

L6564

Hypothesis H24 : SNo (g x x2)

L6565

Hypothesis H25 : SNo (g u v)

L6566

Hypothesis H26 : SNo (g u x2)

L6567

Hypothesis H27 : SNo (g u y)

L6568

Hypothesis H28 : SNo (g x2 z)

L6569

Hypothesis H29 : SNo (g y y2)

L6570

Hypothesis H30 : SNo (g u (g x2 z))

L6571

Hypothesis H31 : SNo (g u (g y y2))

L6572

Hypothesis H32 : SNo (g x2 y2)

L6573

Hypothesis H33 : SNo (g x (g x2 y2))

L6574

Hypothesis H34 : SNo (g x (g x2 z))

L6575

Hypothesis H35 : SNo (g x (g y y2))

L6576

Hypothesis H36 : SNo (g u (g y z) + g x v)

L6577

Hypothesis H37 : SNo (g (g x y) z + g u v)

L6578

Hypothesis H38 : SNo (g u (g x2 y2))

L6579

Hypothesis H39 : SNo (g u (v + g x2 y2))

L6580

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L6581

Theorem. (Conj_mul_SNo_assoc_lem1__3__17)

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

In Proofgold the corresponding term root is 5631a1... and proposition id is 550eab...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__3__38

L6587

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__3__38 TMSqYdGmrpnSAebNhRfhF9U1NUTqAvCAwLC bounty of about 25 bars ***)

L6588

Variable x : set

L6589

Variable y : set

L6590

Variable z : set

L6591

Variable w : set

L6592

Variable u : set

L6593

Variable v : set

L6594

Variable x2 : set

L6595

Variable y2 : set

L6596

Hypothesis H0 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L6597

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

L6598

Hypothesis H2 : SNo x

L6599

Hypothesis H3 : SNo z

L6600

Hypothesis H4 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6601

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6602

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6603

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6604

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6605

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6606

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)))

L6607

Hypothesis H11 : SNo (g x y)

L6608

Hypothesis H12 : SNo (g (g x y) z)

L6609

Hypothesis H13 : u ∈ SNoS_ (SNoLev x)

L6610

Hypothesis H14 : SNo v

L6611

Hypothesis H15 : x2 ∈ SNoS_ (SNoLev y)

L6612

Hypothesis H16 : y2 ∈ SNoS_ (SNoLev z)

L6613

Hypothesis H17 : w = g u (g y z) + g x v + - (g u v)

L6614

Hypothesis H18 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6615

Hypothesis H19 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6616

Hypothesis H20 : SNo u

L6617

Hypothesis H21 : SNo y2

L6618

Hypothesis H22 : SNo (g u (g y z))

L6619

Hypothesis H23 : SNo (g x v)

L6620

Hypothesis H24 : SNo (g x x2)

L6621

Hypothesis H25 : SNo (g u v)

L6622

Hypothesis H26 : SNo (g u x2)

L6623

Hypothesis H27 : SNo (g u y)

L6624

Hypothesis H28 : SNo (g x2 z)

L6625

Hypothesis H29 : SNo (g y y2)

L6626

Hypothesis H30 : SNo (g u (g x2 z))

L6627

Hypothesis H31 : SNo (g u (g y y2))

L6628

Hypothesis H32 : SNo (g x2 y2)

L6629

Hypothesis H33 : SNo (g x (g x2 y2))

L6630

Hypothesis H34 : SNo (g x (g x2 z))

L6631

Hypothesis H35 : SNo (g x (g y y2))

L6632

Hypothesis H36 : SNo (g u (g y z) + g x v)

L6633

Hypothesis H37 : SNo (g (g x y) z + g u v)

L6634

Hypothesis H39 : SNo (g u (v + g x2 y2))

L6635

Hypothesis H40 : SNo (g u (g x2 z + g y y2))

L6636

Theorem. (Conj_mul_SNo_assoc_lem1__3__38)

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

In Proofgold the corresponding term root is d96595... and proposition id is 0aec21...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__4__1

L6642

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__4__1 TMXcAqsTEE1hgtGAT5HjrtMKvK2iNpueWPH bounty of about 25 bars ***)

L6643

Variable x : set

L6644

Variable y : set

L6645

Variable z : set

L6646

Variable w : set

L6647

Variable u : set

L6648

Variable v : set

L6649

Variable x2 : set

L6650

Variable y2 : set

L6651

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

L6652

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

L6653

Hypothesis H3 : SNo x

L6654

Hypothesis H4 : SNo z

L6655

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6656

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6657

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6658

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6659

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6660

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6661

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)))

L6662

Hypothesis H12 : SNo (g x y)

L6663

Hypothesis H13 : SNo (g (g x y) z)

L6664

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6665

Hypothesis H15 : SNo v

L6666

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6667

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6668

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L6669

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6670

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6671

Hypothesis H21 : SNo u

L6672

Hypothesis H22 : SNo y2

L6673

Hypothesis H23 : SNo (g u (g y z))

L6674

Hypothesis H24 : SNo (g x v)

L6675

Hypothesis H25 : SNo (g x x2)

L6676

Hypothesis H26 : SNo (g u v)

L6677

Hypothesis H27 : SNo (g u x2)

L6678

Hypothesis H28 : SNo (g u y)

L6679

Hypothesis H29 : SNo (g x2 z)

L6680

Hypothesis H30 : SNo (g y y2)

L6681

Hypothesis H31 : SNo (g u (g x2 z))

L6682

Hypothesis H32 : SNo (g u (g y y2))

L6683

Hypothesis H33 : SNo (g x2 y2)

L6684

Hypothesis H34 : SNo (g x (g x2 y2))

L6685

Hypothesis H35 : SNo (g x (g x2 z))

L6686

Hypothesis H36 : SNo (g x (g y y2))

L6687

Hypothesis H37 : SNo (g u (g y z) + g x v)

L6688

Hypothesis H38 : SNo (g (g x y) z + g u v)

L6689

Hypothesis H39 : SNo (g u (g x2 y2))

L6690

Hypothesis H40 : SNo (g u (v + g x2 y2))

L6691

Theorem. (Conj_mul_SNo_assoc_lem1__4__1)

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

In Proofgold the corresponding term root is b91589... and proposition id is 9b20f6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__4__8

L6697

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__4__8 TMHqeR25A8wfCNzLRzDFcNrBmLVdymZjKmd bounty of about 25 bars ***)

L6698

Variable x : set

L6699

Variable y : set

L6700

Variable z : set

L6701

Variable w : set

L6702

Variable u : set

L6703

Variable v : set

L6704

Variable x2 : set

L6705

Variable y2 : set

L6706

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

L6707

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

L6708

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

L6709

Hypothesis H3 : SNo x

L6710

Hypothesis H4 : SNo z

L6711

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6712

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6713

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6714

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6715

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6716

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)))

L6717

Hypothesis H12 : SNo (g x y)

L6718

Hypothesis H13 : SNo (g (g x y) z)

L6719

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6720

Hypothesis H15 : SNo v

L6721

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6722

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6723

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L6724

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6725

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6726

Hypothesis H21 : SNo u

L6727

Hypothesis H22 : SNo y2

L6728

Hypothesis H23 : SNo (g u (g y z))

L6729

Hypothesis H24 : SNo (g x v)

L6730

Hypothesis H25 : SNo (g x x2)

L6731

Hypothesis H26 : SNo (g u v)

L6732

Hypothesis H27 : SNo (g u x2)

L6733

Hypothesis H28 : SNo (g u y)

L6734

Hypothesis H29 : SNo (g x2 z)

L6735

Hypothesis H30 : SNo (g y y2)

L6736

Hypothesis H31 : SNo (g u (g x2 z))

L6737

Hypothesis H32 : SNo (g u (g y y2))

L6738

Hypothesis H33 : SNo (g x2 y2)

L6739

Hypothesis H34 : SNo (g x (g x2 y2))

L6740

Hypothesis H35 : SNo (g x (g x2 z))

L6741

Hypothesis H36 : SNo (g x (g y y2))

L6742

Hypothesis H37 : SNo (g u (g y z) + g x v)

L6743

Hypothesis H38 : SNo (g (g x y) z + g u v)

L6744

Hypothesis H39 : SNo (g u (g x2 y2))

L6745

Hypothesis H40 : SNo (g u (v + g x2 y2))

L6746

Theorem. (Conj_mul_SNo_assoc_lem1__4__8)

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

In Proofgold the corresponding term root is d20861... and proposition id is 9f1439...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__4__18

L6752

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__4__18 TMSNQ9v7jdToPn2XowpMzAZxtuxyWQU35tL bounty of about 25 bars ***)

L6753

Variable x : set

L6754

Variable y : set

L6755

Variable z : set

L6756

Variable w : set

L6757

Variable u : set

L6758

Variable v : set

L6759

Variable x2 : set

L6760

Variable y2 : set

L6761

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

L6762

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

L6763

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

L6764

Hypothesis H3 : SNo x

L6765

Hypothesis H4 : SNo z

L6766

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6767

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6768

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6769

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6770

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6771

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6772

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)))

L6773

Hypothesis H12 : SNo (g x y)

L6774

Hypothesis H13 : SNo (g (g x y) z)

L6775

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6776

Hypothesis H15 : SNo v

L6777

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6778

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6779

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6780

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6781

Hypothesis H21 : SNo u

L6782

Hypothesis H22 : SNo y2

L6783

Hypothesis H23 : SNo (g u (g y z))

L6784

Hypothesis H24 : SNo (g x v)

L6785

Hypothesis H25 : SNo (g x x2)

L6786

Hypothesis H26 : SNo (g u v)

L6787

Hypothesis H27 : SNo (g u x2)

L6788

Hypothesis H28 : SNo (g u y)

L6789

Hypothesis H29 : SNo (g x2 z)

L6790

Hypothesis H30 : SNo (g y y2)

L6791

Hypothesis H31 : SNo (g u (g x2 z))

L6792

Hypothesis H32 : SNo (g u (g y y2))

L6793

Hypothesis H33 : SNo (g x2 y2)

L6794

Hypothesis H34 : SNo (g x (g x2 y2))

L6795

Hypothesis H35 : SNo (g x (g x2 z))

L6796

Hypothesis H36 : SNo (g x (g y y2))

L6797

Hypothesis H37 : SNo (g u (g y z) + g x v)

L6798

Hypothesis H38 : SNo (g (g x y) z + g u v)

L6799

Hypothesis H39 : SNo (g u (g x2 y2))

L6800

Hypothesis H40 : SNo (g u (v + g x2 y2))

L6801

Theorem. (Conj_mul_SNo_assoc_lem1__4__18)

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

In Proofgold the corresponding term root is 5efb8c... and proposition id is 41d5b6...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__4__27

L6807

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__4__27 TMRivtAg1Hpj5g8vE8JBaLWPNZ5M99bCppQ bounty of about 25 bars ***)

L6808

Variable x : set

L6809

Variable y : set

L6810

Variable z : set

L6811

Variable w : set

L6812

Variable u : set

L6813

Variable v : set

L6814

Variable x2 : set

L6815

Variable y2 : set

L6816

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

L6817

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

L6818

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

L6819

Hypothesis H3 : SNo x

L6820

Hypothesis H4 : SNo z

L6821

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6822

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6823

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6824

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6825

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6826

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6827

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)))

L6828

Hypothesis H12 : SNo (g x y)

L6829

Hypothesis H13 : SNo (g (g x y) z)

L6830

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6831

Hypothesis H15 : SNo v

L6832

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6833

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6834

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L6835

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6836

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6837

Hypothesis H21 : SNo u

L6838

Hypothesis H22 : SNo y2

L6839

Hypothesis H23 : SNo (g u (g y z))

L6840

Hypothesis H24 : SNo (g x v)

L6841

Hypothesis H25 : SNo (g x x2)

L6842

Hypothesis H26 : SNo (g u v)

L6843

Hypothesis H28 : SNo (g u y)

L6844

Hypothesis H29 : SNo (g x2 z)

L6845

Hypothesis H30 : SNo (g y y2)

L6846

Hypothesis H31 : SNo (g u (g x2 z))

L6847

Hypothesis H32 : SNo (g u (g y y2))

L6848

Hypothesis H33 : SNo (g x2 y2)

L6849

Hypothesis H34 : SNo (g x (g x2 y2))

L6850

Hypothesis H35 : SNo (g x (g x2 z))

L6851

Hypothesis H36 : SNo (g x (g y y2))

L6852

Hypothesis H37 : SNo (g u (g y z) + g x v)

L6853

Hypothesis H38 : SNo (g (g x y) z + g u v)

L6854

Hypothesis H39 : SNo (g u (g x2 y2))

L6855

Hypothesis H40 : SNo (g u (v + g x2 y2))

L6856

Theorem. (Conj_mul_SNo_assoc_lem1__4__27)

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

In Proofgold the corresponding term root is 989325... and proposition id is ada97f...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__6__15

L6862

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__6__15 TMMKp1bkq5dDyp14ALeW9ahjHFkKUqM2nvs bounty of about 25 bars ***)

L6863

Variable x : set

L6864

Variable y : set

L6865

Variable z : set

L6866

Variable w : set

L6867

Variable u : set

L6868

Variable v : set

L6869

Variable x2 : set

L6870

Variable y2 : set

L6871

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

L6872

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

L6873

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

L6874

Hypothesis H3 : SNo x

L6875

Hypothesis H4 : SNo z

L6876

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6877

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6878

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6879

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6880

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6881

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6882

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)))

L6883

Hypothesis H12 : SNo (g x y)

L6884

Hypothesis H13 : SNo (g (g x y) z)

L6885

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6886

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6887

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6888

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L6889

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6890

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6891

Hypothesis H21 : SNo u

L6892

Hypothesis H22 : SNo y2

L6893

Hypothesis H23 : SNo (g u (g y z))

L6894

Hypothesis H24 : SNo (g x v)

L6895

Hypothesis H25 : SNo (g x x2)

L6896

Hypothesis H26 : SNo (g u v)

L6897

Hypothesis H27 : SNo (g u x2)

L6898

Hypothesis H28 : SNo (g u y)

L6899

Hypothesis H29 : SNo (g x2 z)

L6900

Hypothesis H30 : SNo (g y y2)

L6901

Hypothesis H31 : SNo (g u (g x2 z))

L6902

Hypothesis H32 : SNo (g u (g y y2))

L6903

Hypothesis H33 : SNo (g x2 y2)

L6904

Hypothesis H34 : SNo (g x (g x2 y2))

L6905

Hypothesis H35 : SNo (g x (g x2 z))

L6906

Hypothesis H36 : SNo (g x (g y y2))

L6907

Hypothesis H37 : SNo (g u (g y z) + g x v)

L6908

Hypothesis H38 : SNo (g (g x y) z + g u v)

L6909

Hypothesis H39 : SNo (g u (g x2 y2))

L6910

Theorem. (Conj_mul_SNo_assoc_lem1__6__15)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 837556... and proposition id is 28a22e...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__6__31

L6916

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__6__31 TMPXcW5uzUy7EqAgtSNhTECT8qPW2egc1Ur bounty of about 25 bars ***)

L6917

Variable x : set

L6918

Variable y : set

L6919

Variable z : set

L6920

Variable w : set

L6921

Variable u : set

L6922

Variable v : set

L6923

Variable x2 : set

L6924

Variable y2 : set

L6925

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

L6926

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

L6927

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

L6928

Hypothesis H3 : SNo x

L6929

Hypothesis H4 : SNo z

L6930

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6931

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6932

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6933

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6934

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6935

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6936

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)))

L6937

Hypothesis H12 : SNo (g x y)

L6938

Hypothesis H13 : SNo (g (g x y) z)

L6939

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6940

Hypothesis H15 : SNo v

L6941

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6942

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6943

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L6944

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6945

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L6946

Hypothesis H21 : SNo u

L6947

Hypothesis H22 : SNo y2

L6948

Hypothesis H23 : SNo (g u (g y z))

L6949

Hypothesis H24 : SNo (g x v)

L6950

Hypothesis H25 : SNo (g x x2)

L6951

Hypothesis H26 : SNo (g u v)

L6952

Hypothesis H27 : SNo (g u x2)

L6953

Hypothesis H28 : SNo (g u y)

L6954

Hypothesis H29 : SNo (g x2 z)

L6955

Hypothesis H30 : SNo (g y y2)

L6956

Hypothesis H32 : SNo (g u (g y y2))

L6957

Hypothesis H33 : SNo (g x2 y2)

L6958

Hypothesis H34 : SNo (g x (g x2 y2))

L6959

Hypothesis H35 : SNo (g x (g x2 z))

L6960

Hypothesis H36 : SNo (g x (g y y2))

L6961

Hypothesis H37 : SNo (g u (g y z) + g x v)

L6962

Hypothesis H38 : SNo (g (g x y) z + g u v)

L6963

Hypothesis H39 : SNo (g u (g x2 y2))

L6964

Theorem. (Conj_mul_SNo_assoc_lem1__6__31)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is c8be30... and proposition id is 8447c7...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__6__39

L6970

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__6__39 TMTWYzUV5UDpy1hKuhnZarfTwo7KBRde5wJ bounty of about 25 bars ***)

L6971

Variable x : set

L6972

Variable y : set

L6973

Variable z : set

L6974

Variable w : set

L6975

Variable u : set

L6976

Variable v : set

L6977

Variable x2 : set

L6978

Variable y2 : set

L6979

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

L6980

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

L6981

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

L6982

Hypothesis H3 : SNo x

L6983

Hypothesis H4 : SNo z

L6984

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L6985

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L6986

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L6987

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L6988

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L6989

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L6990

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)))

L6991

Hypothesis H12 : SNo (g x y)

L6992

Hypothesis H13 : SNo (g (g x y) z)

L6993

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L6994

Hypothesis H15 : SNo v

L6995

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L6996

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L6997

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L6998

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L6999

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7000

Hypothesis H21 : SNo u

L7001

Hypothesis H22 : SNo y2

L7002

Hypothesis H23 : SNo (g u (g y z))

L7003

Hypothesis H24 : SNo (g x v)

L7004

Hypothesis H25 : SNo (g x x2)

L7005

Hypothesis H26 : SNo (g u v)

L7006

Hypothesis H27 : SNo (g u x2)

L7007

Hypothesis H28 : SNo (g u y)

L7008

Hypothesis H29 : SNo (g x2 z)

L7009

Hypothesis H30 : SNo (g y y2)

L7010

Hypothesis H31 : SNo (g u (g x2 z))

L7011

Hypothesis H32 : SNo (g u (g y y2))

L7012

Hypothesis H33 : SNo (g x2 y2)

L7013

Hypothesis H34 : SNo (g x (g x2 y2))

L7014

Hypothesis H35 : SNo (g x (g x2 z))

L7015

Hypothesis H36 : SNo (g x (g y y2))

L7016

Hypothesis H37 : SNo (g u (g y z) + g x v)

L7017

Hypothesis H38 : SNo (g (g x y) z + g u v)

L7018

Theorem. (Conj_mul_SNo_assoc_lem1__6__39)

SNo (v + g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 0e48cb... and proposition id is 8b5844...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__9__22

L7024

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__9__22 TML5UvhA1EB19XK4JVreoGD2HLShnGjvJ1L bounty of about 25 bars ***)

L7025

Variable x : set

L7026

Variable y : set

L7027

Variable z : set

L7028

Variable w : set

L7029

Variable u : set

L7030

Variable v : set

L7031

Variable x2 : set

L7032

Variable y2 : set

L7033

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

L7034

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

L7035

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

L7036

Hypothesis H3 : SNo x

L7037

Hypothesis H4 : SNo z

L7038

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7039

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7040

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7041

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7042

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7043

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7044

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)))

L7045

Hypothesis H12 : SNo (g x y)

L7046

Hypothesis H13 : SNo (g (g x y) z)

L7047

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7048

Hypothesis H15 : SNo v

L7049

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7050

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7051

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7052

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7053

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7054

Hypothesis H21 : SNo u

L7055

Hypothesis H23 : SNo (g u (g y z))

L7056

Hypothesis H24 : SNo (g x v)

L7057

Hypothesis H25 : SNo (g x x2)

L7058

Hypothesis H26 : SNo (g u v)

L7059

Hypothesis H27 : SNo (g u x2)

L7060

Hypothesis H28 : SNo (g u y)

L7061

Hypothesis H29 : SNo (g x2 z)

L7062

Hypothesis H30 : SNo (g y y2)

L7063

Hypothesis H31 : SNo (g u (g x2 z))

L7064

Hypothesis H32 : SNo (g u (g y y2))

L7065

Hypothesis H33 : SNo (g x2 y2)

L7066

Hypothesis H34 : SNo (g x (g x2 y2))

L7067

Hypothesis H35 : SNo (g x (g x2 z))

L7068

Hypothesis H36 : SNo (g x (g y y2))

L7069

Theorem. (Conj_mul_SNo_assoc_lem1__9__22)

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

In Proofgold the corresponding term root is 8e37c7... and proposition id is d619e2...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__10__14

L7075

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__10__14 TMciT5MQuzNPVkBa3VvEFY6mhkvQM44PZuM bounty of about 25 bars ***)

L7076

Variable x : set

L7077

Variable y : set

L7078

Variable z : set

L7079

Variable w : set

L7080

Variable u : set

L7081

Variable v : set

L7082

Variable x2 : set

L7083

Variable y2 : set

L7084

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

L7085

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

L7086

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

L7087

Hypothesis H3 : SNo x

L7088

Hypothesis H4 : SNo z

L7089

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7090

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7091

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7092

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7093

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7094

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7095

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)))

L7096

Hypothesis H12 : SNo (g x y)

L7097

Hypothesis H13 : SNo (g (g x y) z)

L7098

Hypothesis H15 : SNo v

L7099

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7100

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7101

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7102

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7103

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7104

Hypothesis H21 : SNo u

L7105

Hypothesis H22 : SNo y2

L7106

Hypothesis H23 : SNo (g u (g y z))

L7107

Hypothesis H24 : SNo (g x v)

L7108

Hypothesis H25 : SNo (g x x2)

L7109

Hypothesis H26 : SNo (g u v)

L7110

Hypothesis H27 : SNo (g u x2)

L7111

Hypothesis H28 : SNo (g u y)

L7112

Hypothesis H29 : SNo (g x2 z)

L7113

Hypothesis H30 : SNo (g y y2)

L7114

Hypothesis H31 : SNo (g u (g x2 z))

L7115

Hypothesis H32 : SNo (g u (g y y2))

L7116

Hypothesis H33 : SNo (g x2 y2)

L7117

Hypothesis H34 : SNo (g x (g x2 y2))

L7118

Hypothesis H35 : SNo (g x (g x2 z))

L7119

Theorem. (Conj_mul_SNo_assoc_lem1__10__14)

SNo (g x (g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 6b314d... and proposition id is e8373d...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__10__16

L7125

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__10__16 TMQQUwReS53D6Ao3Pua1pAeLJXeWQanM8Uf bounty of about 25 bars ***)

L7126

Variable x : set

L7127

Variable y : set

L7128

Variable z : set

L7129

Variable w : set

L7130

Variable u : set

L7131

Variable v : set

L7132

Variable x2 : set

L7133

Variable y2 : set

L7134

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

L7135

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

L7136

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

L7137

Hypothesis H3 : SNo x

L7138

Hypothesis H4 : SNo z

L7139

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7140

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7141

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7142

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7143

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7144

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7145

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)))

L7146

Hypothesis H12 : SNo (g x y)

L7147

Hypothesis H13 : SNo (g (g x y) z)

L7148

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7149

Hypothesis H15 : SNo v

L7150

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7151

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7152

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7153

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7154

Hypothesis H21 : SNo u

L7155

Hypothesis H22 : SNo y2

L7156

Hypothesis H23 : SNo (g u (g y z))

L7157

Hypothesis H24 : SNo (g x v)

L7158

Hypothesis H25 : SNo (g x x2)

L7159

Hypothesis H26 : SNo (g u v)

L7160

Hypothesis H27 : SNo (g u x2)

L7161

Hypothesis H28 : SNo (g u y)

L7162

Hypothesis H29 : SNo (g x2 z)

L7163

Hypothesis H30 : SNo (g y y2)

L7164

Hypothesis H31 : SNo (g u (g x2 z))

L7165

Hypothesis H32 : SNo (g u (g y y2))

L7166

Hypothesis H33 : SNo (g x2 y2)

L7167

Hypothesis H34 : SNo (g x (g x2 y2))

L7168

Hypothesis H35 : SNo (g x (g x2 z))

L7169

Theorem. (Conj_mul_SNo_assoc_lem1__10__16)

SNo (g x (g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is d80850... and proposition id is 1cf946...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__10__19

L7175

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__10__19 TMRVGx7eJ9f36S1ZzNyy8Y1R5BCct4CBbU6 bounty of about 25 bars ***)

L7176

Variable x : set

L7177

Variable y : set

L7178

Variable z : set

L7179

Variable w : set

L7180

Variable u : set

L7181

Variable v : set

L7182

Variable x2 : set

L7183

Variable y2 : set

L7184

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

L7185

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

L7186

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

L7187

Hypothesis H3 : SNo x

L7188

Hypothesis H4 : SNo z

L7189

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7190

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7191

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7192

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7193

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7194

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7195

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)))

L7196

Hypothesis H12 : SNo (g x y)

L7197

Hypothesis H13 : SNo (g (g x y) z)

L7198

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7199

Hypothesis H15 : SNo v

L7200

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7201

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7202

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7203

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7204

Hypothesis H21 : SNo u

L7205

Hypothesis H22 : SNo y2

L7206

Hypothesis H23 : SNo (g u (g y z))

L7207

Hypothesis H24 : SNo (g x v)

L7208

Hypothesis H25 : SNo (g x x2)

L7209

Hypothesis H26 : SNo (g u v)

L7210

Hypothesis H27 : SNo (g u x2)

L7211

Hypothesis H28 : SNo (g u y)

L7212

Hypothesis H29 : SNo (g x2 z)

L7213

Hypothesis H30 : SNo (g y y2)

L7214

Hypothesis H31 : SNo (g u (g x2 z))

L7215

Hypothesis H32 : SNo (g u (g y y2))

L7216

Hypothesis H33 : SNo (g x2 y2)

L7217

Hypothesis H34 : SNo (g x (g x2 y2))

L7218

Hypothesis H35 : SNo (g x (g x2 z))

L7219

Theorem. (Conj_mul_SNo_assoc_lem1__10__19)

SNo (g x (g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 0bdc1f... and proposition id is 389e35...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__10__26

L7225

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__10__26 TMWFSt8CQsyePtHHgC3w6gXt9FkTq72zq6n bounty of about 25 bars ***)

L7226

Variable x : set

L7227

Variable y : set

L7228

Variable z : set

L7229

Variable w : set

L7230

Variable u : set

L7231

Variable v : set

L7232

Variable x2 : set

L7233

Variable y2 : set

L7234

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

L7235

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

L7236

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

L7237

Hypothesis H3 : SNo x

L7238

Hypothesis H4 : SNo z

L7239

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7240

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7241

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7242

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7243

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7244

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7245

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)))

L7246

Hypothesis H12 : SNo (g x y)

L7247

Hypothesis H13 : SNo (g (g x y) z)

L7248

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7249

Hypothesis H15 : SNo v

L7250

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7251

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7252

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7253

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7254

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7255

Hypothesis H21 : SNo u

L7256

Hypothesis H22 : SNo y2

L7257

Hypothesis H23 : SNo (g u (g y z))

L7258

Hypothesis H24 : SNo (g x v)

L7259

Hypothesis H25 : SNo (g x x2)

L7260

Hypothesis H27 : SNo (g u x2)

L7261

Hypothesis H28 : SNo (g u y)

L7262

Hypothesis H29 : SNo (g x2 z)

L7263

Hypothesis H30 : SNo (g y y2)

L7264

Hypothesis H31 : SNo (g u (g x2 z))

L7265

Hypothesis H32 : SNo (g u (g y y2))

L7266

Hypothesis H33 : SNo (g x2 y2)

L7267

Hypothesis H34 : SNo (g x (g x2 y2))

L7268

Hypothesis H35 : SNo (g x (g x2 z))

L7269

Theorem. (Conj_mul_SNo_assoc_lem1__10__26)

SNo (g x (g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 667a36... and proposition id is 5732a3...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__12__2

L7275

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__12__2 TMZo1YdifN3e5enp4ydBHys7i9ghGmkcy3N bounty of about 25 bars ***)

L7276

Variable x : set

L7277

Variable y : set

L7278

Variable z : set

L7279

Variable w : set

L7280

Variable u : set

L7281

Variable v : set

L7282

Variable x2 : set

L7283

Variable y2 : set

L7284

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

L7285

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

L7286

Hypothesis H3 : SNo x

L7287

Hypothesis H4 : SNo z

L7288

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7289

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7290

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7291

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7292

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7293

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7294

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)))

L7295

Hypothesis H12 : SNo (g x y)

L7296

Hypothesis H13 : SNo (g (g x y) z)

L7297

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7298

Hypothesis H15 : SNo v

L7299

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7300

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7301

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7302

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7303

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7304

Hypothesis H21 : SNo u

L7305

Hypothesis H22 : SNo y2

L7306

Hypothesis H23 : SNo (g u (g y z))

L7307

Hypothesis H24 : SNo (g x v)

L7308

Hypothesis H25 : SNo (g x x2)

L7309

Hypothesis H26 : SNo (g u v)

L7310

Hypothesis H27 : SNo (g u x2)

L7311

Hypothesis H28 : SNo (g u y)

L7312

Hypothesis H29 : SNo (g x2 z)

L7313

Hypothesis H30 : SNo (g y y2)

L7314

Hypothesis H31 : SNo (g u (g x2 z))

L7315

Hypothesis H32 : SNo (g u (g y y2))

L7316

Hypothesis H33 : SNo (g x2 y2)

L7317

Theorem. (Conj_mul_SNo_assoc_lem1__12__2)

SNo (g x (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is b98340... and proposition id is 694244...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__12__16

L7323

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__12__16 TMPXwsxij2gSSDac57YbY16Wika8rQ18gTL bounty of about 25 bars ***)

L7324

Variable x : set

L7325

Variable y : set

L7326

Variable z : set

L7327

Variable w : set

L7328

Variable u : set

L7329

Variable v : set

L7330

Variable x2 : set

L7331

Variable y2 : set

L7332

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

L7333

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

L7334

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

L7335

Hypothesis H3 : SNo x

L7336

Hypothesis H4 : SNo z

L7337

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7338

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7339

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7340

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7341

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7342

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7343

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)))

L7344

Hypothesis H12 : SNo (g x y)

L7345

Hypothesis H13 : SNo (g (g x y) z)

L7346

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7347

Hypothesis H15 : SNo v

L7348

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7349

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7350

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7351

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7352

Hypothesis H21 : SNo u

L7353

Hypothesis H22 : SNo y2

L7354

Hypothesis H23 : SNo (g u (g y z))

L7355

Hypothesis H24 : SNo (g x v)

L7356

Hypothesis H25 : SNo (g x x2)

L7357

Hypothesis H26 : SNo (g u v)

L7358

Hypothesis H27 : SNo (g u x2)

L7359

Hypothesis H28 : SNo (g u y)

L7360

Hypothesis H29 : SNo (g x2 z)

L7361

Hypothesis H30 : SNo (g y y2)

L7362

Hypothesis H31 : SNo (g u (g x2 z))

L7363

Hypothesis H32 : SNo (g u (g y y2))

L7364

Hypothesis H33 : SNo (g x2 y2)

L7365

Theorem. (Conj_mul_SNo_assoc_lem1__12__16)

SNo (g x (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 785ab5... and proposition id is 30c723...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__12__17

L7371

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__12__17 TMVnHxQQNK4PGT3sHgK5wJFtaB6LWs1aEWx bounty of about 25 bars ***)

L7372

Variable x : set

L7373

Variable y : set

L7374

Variable z : set

L7375

Variable w : set

L7376

Variable u : set

L7377

Variable v : set

L7378

Variable x2 : set

L7379

Variable y2 : set

L7380

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

L7381

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

L7382

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

L7383

Hypothesis H3 : SNo x

L7384

Hypothesis H4 : SNo z

L7385

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7386

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7387

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7388

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7389

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7390

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7391

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)))

L7392

Hypothesis H12 : SNo (g x y)

L7393

Hypothesis H13 : SNo (g (g x y) z)

L7394

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7395

Hypothesis H15 : SNo v

L7396

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7397

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7398

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7399

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7400

Hypothesis H21 : SNo u

L7401

Hypothesis H22 : SNo y2

L7402

Hypothesis H23 : SNo (g u (g y z))

L7403

Hypothesis H24 : SNo (g x v)

L7404

Hypothesis H25 : SNo (g x x2)

L7405

Hypothesis H26 : SNo (g u v)

L7406

Hypothesis H27 : SNo (g u x2)

L7407

Hypothesis H28 : SNo (g u y)

L7408

Hypothesis H29 : SNo (g x2 z)

L7409

Hypothesis H30 : SNo (g y y2)

L7410

Hypothesis H31 : SNo (g u (g x2 z))

L7411

Hypothesis H32 : SNo (g u (g y y2))

L7412

Hypothesis H33 : SNo (g x2 y2)

L7413

Theorem. (Conj_mul_SNo_assoc_lem1__12__17)

SNo (g x (g x2 y2)) → w < g (g x y) z

In Proofgold the corresponding term root is f61890... and proposition id is 50c2e8...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__13__6

L7419

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__13__6 TMMq2w8qWm3wmZmQdsNFjkpwHhP7uU6Vc8V bounty of about 25 bars ***)

L7420

Variable x : set

L7421

Variable y : set

L7422

Variable z : set

L7423

Variable w : set

L7424

Variable u : set

L7425

Variable v : set

L7426

Variable x2 : set

L7427

Variable y2 : set

L7428

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

L7429

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

L7430

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

L7431

Hypothesis H3 : SNo x

L7432

Hypothesis H4 : SNo z

L7433

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7434

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7435

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7436

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7437

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7438

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)))

L7439

Hypothesis H12 : SNo (g x y)

L7440

Hypothesis H13 : SNo (g (g x y) z)

L7441

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7442

Hypothesis H15 : SNo v

L7443

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7444

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7445

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7446

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7447

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7448

Hypothesis H21 : SNo u

L7449

Hypothesis H22 : SNo x2

L7450

Hypothesis H23 : SNo y2

L7451

Hypothesis H24 : SNo (g u (g y z))

L7452

Hypothesis H25 : SNo (g x v)

L7453

Hypothesis H26 : SNo (g x x2)

L7454

Hypothesis H27 : SNo (g u v)

L7455

Hypothesis H28 : SNo (g u x2)

L7456

Hypothesis H29 : SNo (g u y)

L7457

Hypothesis H30 : SNo (g x2 z)

L7458

Hypothesis H31 : SNo (g y y2)

L7459

Hypothesis H32 : SNo (g u (g x2 z))

L7460

Hypothesis H33 : SNo (g u (g y y2))

L7461

Theorem. (Conj_mul_SNo_assoc_lem1__13__6)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 5e70bc... and proposition id is ff8915...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__13__12

L7467

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__13__12 TMa3hMPDPyKTYUenuDmjD9MyWFLydGTRNaD bounty of about 25 bars ***)

L7468

Variable x : set

L7469

Variable y : set

L7470

Variable z : set

L7471

Variable w : set

L7472

Variable u : set

L7473

Variable v : set

L7474

Variable x2 : set

L7475

Variable y2 : set

L7476

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

L7477

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

L7478

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

L7479

Hypothesis H3 : SNo x

L7480

Hypothesis H4 : SNo z

L7481

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7482

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7483

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7484

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7485

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7486

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7487

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)))

L7488

Hypothesis H13 : SNo (g (g x y) z)

L7489

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7490

Hypothesis H15 : SNo v

L7491

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7492

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7493

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7494

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7495

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7496

Hypothesis H21 : SNo u

L7497

Hypothesis H22 : SNo x2

L7498

Hypothesis H23 : SNo y2

L7499

Hypothesis H24 : SNo (g u (g y z))

L7500

Hypothesis H25 : SNo (g x v)

L7501

Hypothesis H26 : SNo (g x x2)

L7502

Hypothesis H27 : SNo (g u v)

L7503

Hypothesis H28 : SNo (g u x2)

L7504

Hypothesis H29 : SNo (g u y)

L7505

Hypothesis H30 : SNo (g x2 z)

L7506

Hypothesis H31 : SNo (g y y2)

L7507

Hypothesis H32 : SNo (g u (g x2 z))

L7508

Hypothesis H33 : SNo (g u (g y y2))

L7509

Theorem. (Conj_mul_SNo_assoc_lem1__13__12)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 00b43f... and proposition id is 654c9a...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__13__26

L7515

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__13__26 TMZ1iaAArizmmL56DcJgDa6t1M26M8Bgzmi bounty of about 25 bars ***)

L7516

Variable x : set

L7517

Variable y : set

L7518

Variable z : set

L7519

Variable w : set

L7520

Variable u : set

L7521

Variable v : set

L7522

Variable x2 : set

L7523

Variable y2 : set

L7524

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

L7525

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

L7526

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

L7527

Hypothesis H3 : SNo x

L7528

Hypothesis H4 : SNo z

L7529

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7530

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7531

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7532

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7533

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7534

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7535

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)))

L7536

Hypothesis H12 : SNo (g x y)

L7537

Hypothesis H13 : SNo (g (g x y) z)

L7538

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7539

Hypothesis H15 : SNo v

L7540

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7541

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7542

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7543

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7544

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7545

Hypothesis H21 : SNo u

L7546

Hypothesis H22 : SNo x2

L7547

Hypothesis H23 : SNo y2

L7548

Hypothesis H24 : SNo (g u (g y z))

L7549

Hypothesis H25 : SNo (g x v)

L7550

Hypothesis H27 : SNo (g u v)

L7551

Hypothesis H28 : SNo (g u x2)

L7552

Hypothesis H29 : SNo (g u y)

L7553

Hypothesis H30 : SNo (g x2 z)

L7554

Hypothesis H31 : SNo (g y y2)

L7555

Hypothesis H32 : SNo (g u (g x2 z))

L7556

Hypothesis H33 : SNo (g u (g y y2))

L7557

Theorem. (Conj_mul_SNo_assoc_lem1__13__26)

SNo (g x2 y2) → w < g (g x y) z

In Proofgold the corresponding term root is 4ef3de... and proposition id is c4f3d9...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__14__12

L7563

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__14__12 TMTrG48ZSckWqGvwCzWNNEhyG34pswmW1KS bounty of about 25 bars ***)

L7564

Variable x : set

L7565

Variable y : set

L7566

Variable z : set

L7567

Variable w : set

L7568

Variable u : set

L7569

Variable v : set

L7570

Variable x2 : set

L7571

Variable y2 : set

L7572

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

L7573

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

L7574

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

L7575

Hypothesis H3 : SNo x

L7576

Hypothesis H4 : SNo z

L7577

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7578

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7579

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7580

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7581

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7582

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7583

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)))

L7584

Hypothesis H13 : SNo (g (g x y) z)

L7585

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7586

Hypothesis H15 : SNo v

L7587

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7588

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7589

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7590

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7591

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7592

Hypothesis H21 : SNo u

L7593

Hypothesis H22 : SNo x2

L7594

Hypothesis H23 : SNo y2

L7595

Hypothesis H24 : SNo (g u (g y z))

L7596

Hypothesis H25 : SNo (g x v)

L7597

Hypothesis H26 : SNo (g x x2)

L7598

Hypothesis H27 : SNo (g u v)

L7599

Hypothesis H28 : SNo (g u x2)

L7600

Hypothesis H29 : SNo (g u y)

L7601

Hypothesis H30 : SNo (g x2 z)

L7602

Hypothesis H31 : SNo (g y y2)

L7603

Hypothesis H32 : SNo (g u (g x2 z))

L7604

Theorem. (Conj_mul_SNo_assoc_lem1__14__12)

SNo (g u (g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 436614... and proposition id is f8aa09...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__14__21

L7610

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__14__21 TMFpz8ZrD1CgD8aTy4QqkjTACqbqDxvi3AN bounty of about 25 bars ***)

L7611

Variable x : set

L7612

Variable y : set

L7613

Variable z : set

L7614

Variable w : set

L7615

Variable u : set

L7616

Variable v : set

L7617

Variable x2 : set

L7618

Variable y2 : set

L7619

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

L7620

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

L7621

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

L7622

Hypothesis H3 : SNo x

L7623

Hypothesis H4 : SNo z

L7624

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7625

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7626

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7627

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7628

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7629

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7630

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)))

L7631

Hypothesis H12 : SNo (g x y)

L7632

Hypothesis H13 : SNo (g (g x y) z)

L7633

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7634

Hypothesis H15 : SNo v

L7635

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7636

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7637

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7638

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7639

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7640

Hypothesis H22 : SNo x2

L7641

Hypothesis H23 : SNo y2

L7642

Hypothesis H24 : SNo (g u (g y z))

L7643

Hypothesis H25 : SNo (g x v)

L7644

Hypothesis H26 : SNo (g x x2)

L7645

Hypothesis H27 : SNo (g u v)

L7646

Hypothesis H28 : SNo (g u x2)

L7647

Hypothesis H29 : SNo (g u y)

L7648

Hypothesis H30 : SNo (g x2 z)

L7649

Hypothesis H31 : SNo (g y y2)

L7650

Hypothesis H32 : SNo (g u (g x2 z))

L7651

Theorem. (Conj_mul_SNo_assoc_lem1__14__21)

SNo (g u (g y y2)) → w < g (g x y) z

In Proofgold the corresponding term root is 14c416... and proposition id is a56f90...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__15__6

L7657

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__15__6 TMHEBdM2iQR8sxxGuChA4vnQ52YjKV8QT9x bounty of about 25 bars ***)

L7658

Variable x : set

L7659

Variable y : set

L7660

Variable z : set

L7661

Variable w : set

L7662

Variable u : set

L7663

Variable v : set

L7664

Variable x2 : set

L7665

Variable y2 : set

L7666

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

L7667

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

L7668

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

L7669

Hypothesis H3 : SNo x

L7670

Hypothesis H4 : SNo z

L7671

Hypothesis H5 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7672

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7673

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7674

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7675

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7676

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)))

L7677

Hypothesis H12 : SNo (g x y)

L7678

Hypothesis H13 : SNo (g (g x y) z)

L7679

Hypothesis H14 : u ∈ SNoS_ (SNoLev x)

L7680

Hypothesis H15 : SNo v

L7681

Hypothesis H16 : x2 ∈ SNoS_ (SNoLev y)

L7682

Hypothesis H17 : y2 ∈ SNoS_ (SNoLev z)

L7683

Hypothesis H18 : w = g u (g y z) + g x v + - (g u v)

L7684

Hypothesis H19 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7685

Hypothesis H20 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7686

Hypothesis H21 : SNo u

L7687

Hypothesis H22 : SNo x2

L7688

Hypothesis H23 : SNo y2

L7689

Hypothesis H24 : SNo (g u (g y z))

L7690

Hypothesis H25 : SNo (g x v)

L7691

Hypothesis H26 : SNo (g x x2)

L7692

Hypothesis H27 : SNo (g u v)

L7693

Hypothesis H28 : SNo (g u x2)

L7694

Hypothesis H29 : SNo (g u y)

L7695

Hypothesis H30 : SNo (g x2 z)

L7696

Hypothesis H31 : SNo (g y y2)

L7697

Theorem. (Conj_mul_SNo_assoc_lem1__15__6)

SNo (g u (g x2 z)) → w < g (g x y) z

In Proofgold the corresponding term root is 4dbf33... and proposition id is 2d9633...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__17__0

L7703

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__17__0 TMWedQg5CTC7ex2qFLyh3WZun2S5CvtZppW 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 H1 : (∀z2 : set, ∀w2 : set, ∀u2 : set, SNo z2 → SNo w2 → SNo u2 → g z2 (w2 + u2) = g z2 w2 + g z2 u2)

L7713

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

L7714

Hypothesis H3 : SNo x

L7715

Hypothesis H4 : SNo y

L7716

Hypothesis H5 : SNo z

L7717

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7718

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7719

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7720

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7721

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7722

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7723

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)))

L7724

Hypothesis H13 : SNo (g x y)

L7725

Hypothesis H14 : SNo (g (g x y) z)

L7726

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7727

Hypothesis H16 : SNo v

L7728

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7729

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7730

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7731

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7732

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7733

Hypothesis H22 : SNo u

L7734

Hypothesis H23 : SNo x2

L7735

Hypothesis H24 : SNo y2

L7736

Hypothesis H25 : SNo (g u (g y z))

L7737

Hypothesis H26 : SNo (g x v)

L7738

Hypothesis H27 : SNo (g x x2)

L7739

Hypothesis H28 : SNo (g u v)

L7740

Hypothesis H29 : SNo (g u x2)

L7741

Hypothesis H30 : SNo (g u y)

L7742

Theorem. (Conj_mul_SNo_assoc_lem1__17__0)

SNo (g x2 z) → w < g (g x y) z

In Proofgold the corresponding term root is 6a847f... and proposition id is 5ee3c5...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__17__2

L7748

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__17__2 TMW4cHKhGeVSoDs8o9MKboxwjgkF7HcmLKi bounty of about 25 bars ***)

L7749

Variable x : set

L7750

Variable y : set

L7751

Variable z : set

L7752

Variable w : set

L7753

Variable u : set

L7754

Variable v : set

L7755

Variable x2 : set

L7756

Variable y2 : set

L7757

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

L7758

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

L7759

Hypothesis H3 : SNo x

L7760

Hypothesis H4 : SNo y

L7761

Hypothesis H5 : SNo z

L7762

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7763

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7764

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7765

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7766

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7767

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7768

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)))

L7769

Hypothesis H13 : SNo (g x y)

L7770

Hypothesis H14 : SNo (g (g x y) z)

L7771

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7772

Hypothesis H16 : SNo v

L7773

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7774

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7775

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7776

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7777

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7778

Hypothesis H22 : SNo u

L7779

Hypothesis H23 : SNo x2

L7780

Hypothesis H24 : SNo y2

L7781

Hypothesis H25 : SNo (g u (g y z))

L7782

Hypothesis H26 : SNo (g x v)

L7783

Hypothesis H27 : SNo (g x x2)

L7784

Hypothesis H28 : SNo (g u v)

L7785

Hypothesis H29 : SNo (g u x2)

L7786

Hypothesis H30 : SNo (g u y)

L7787

Theorem. (Conj_mul_SNo_assoc_lem1__17__2)

SNo (g x2 z) → w < g (g x y) z

In Proofgold the corresponding term root is d1a0d4... and proposition id is cdff02...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__17__14

L7793

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__17__14 TMWJwMJWR15yeki7TxMRQhNDYjo5YENFH1V bounty of about 25 bars ***)

L7794

Variable x : set

L7795

Variable y : set

L7796

Variable z : set

L7797

Variable w : set

L7798

Variable u : set

L7799

Variable v : set

L7800

Variable x2 : set

L7801

Variable y2 : set

L7802

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

L7803

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

L7804

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

L7805

Hypothesis H3 : SNo x

L7806

Hypothesis H4 : SNo y

L7807

Hypothesis H5 : SNo z

L7808

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7809

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7810

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7811

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7812

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7813

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7814

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)))

L7815

Hypothesis H13 : SNo (g x y)

L7816

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7817

Hypothesis H16 : SNo v

L7818

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7819

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7820

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7821

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7822

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7823

Hypothesis H22 : SNo u

L7824

Hypothesis H23 : SNo x2

L7825

Hypothesis H24 : SNo y2

L7826

Hypothesis H25 : SNo (g u (g y z))

L7827

Hypothesis H26 : SNo (g x v)

L7828

Hypothesis H27 : SNo (g x x2)

L7829

Hypothesis H28 : SNo (g u v)

L7830

Hypothesis H29 : SNo (g u x2)

L7831

Hypothesis H30 : SNo (g u y)

L7832

Theorem. (Conj_mul_SNo_assoc_lem1__17__14)

SNo (g x2 z) → w < g (g x y) z

In Proofgold the corresponding term root is fc9919... and proposition id is 074dd1...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__18__24

L7838

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__18__24 TMYPtSgqDEFm4NbTPaDe1sr88v4ZMUEHije bounty of about 25 bars ***)

L7839

Variable x : set

L7840

Variable y : set

L7841

Variable z : set

L7842

Variable w : set

L7843

Variable u : set

L7844

Variable v : set

L7845

Variable x2 : set

L7846

Variable y2 : set

L7847

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

L7848

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

L7849

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

L7850

Hypothesis H3 : SNo x

L7851

Hypothesis H4 : SNo y

L7852

Hypothesis H5 : SNo z

L7853

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7854

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7855

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7856

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7857

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7858

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7859

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)))

L7860

Hypothesis H13 : SNo (g x y)

L7861

Hypothesis H14 : SNo (g (g x y) z)

L7862

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7863

Hypothesis H16 : SNo v

L7864

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7865

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7866

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7867

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7868

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7869

Hypothesis H22 : SNo u

L7870

Hypothesis H23 : SNo x2

L7871

Hypothesis H25 : SNo (g u (g y z))

L7872

Hypothesis H26 : SNo (g x v)

L7873

Hypothesis H27 : SNo (g x x2)

L7874

Hypothesis H28 : SNo (g u v)

L7875

Hypothesis H29 : SNo (g u x2)

L7876

Theorem. (Conj_mul_SNo_assoc_lem1__18__24)

SNo (g u y) → w < g (g x y) z

In Proofgold the corresponding term root is 1672a6... and proposition id is 11bb64...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__18__27

L7882

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__18__27 TMR3Xczm7LHwMZp8fFPDwDaHkeHYkCdxsPP bounty of about 25 bars ***)

L7883

Variable x : set

L7884

Variable y : set

L7885

Variable z : set

L7886

Variable w : set

L7887

Variable u : set

L7888

Variable v : set

L7889

Variable x2 : set

L7890

Variable y2 : set

L7891

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

L7892

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

L7893

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

L7894

Hypothesis H3 : SNo x

L7895

Hypothesis H4 : SNo y

L7896

Hypothesis H5 : SNo z

L7897

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7898

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7899

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7900

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7901

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7902

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7903

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)))

L7904

Hypothesis H13 : SNo (g x y)

L7905

Hypothesis H14 : SNo (g (g x y) z)

L7906

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7907

Hypothesis H16 : SNo v

L7908

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7909

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7910

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7911

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7912

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7913

Hypothesis H22 : SNo u

L7914

Hypothesis H23 : SNo x2

L7915

Hypothesis H24 : SNo y2

L7916

Hypothesis H25 : SNo (g u (g y z))

L7917

Hypothesis H26 : SNo (g x v)

L7918

Hypothesis H28 : SNo (g u v)

L7919

Hypothesis H29 : SNo (g u x2)

L7920

Theorem. (Conj_mul_SNo_assoc_lem1__18__27)

SNo (g u y) → w < g (g x y) z

In Proofgold the corresponding term root is 1f06e7... and proposition id is 02c8ba...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__19__13

L7926

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__19__13 TMbQw1uPYngB8ZEjaNZREXfTr4Aj8YNm2My bounty of about 25 bars ***)

L7927

Variable x : set

L7928

Variable y : set

L7929

Variable z : set

L7930

Variable w : set

L7931

Variable u : set

L7932

Variable v : set

L7933

Variable x2 : set

L7934

Variable y2 : set

L7935

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

L7936

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

L7937

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

L7938

Hypothesis H3 : SNo x

L7939

Hypothesis H4 : SNo y

L7940

Hypothesis H5 : SNo z

L7941

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7942

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7943

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7944

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7945

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7946

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7947

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)))

L7948

Hypothesis H14 : SNo (g (g x y) z)

L7949

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7950

Hypothesis H16 : SNo v

L7951

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7952

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7953

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7954

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7955

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7956

Hypothesis H22 : SNo u

L7957

Hypothesis H23 : SNo x2

L7958

Hypothesis H24 : SNo y2

L7959

Hypothesis H25 : SNo (g u (g y z))

L7960

Hypothesis H26 : SNo (g x v)

L7961

Hypothesis H27 : SNo (g x x2)

L7962

Hypothesis H28 : SNo (g u v)

L7963

Theorem. (Conj_mul_SNo_assoc_lem1__19__13)

SNo (g u x2) → w < g (g x y) z

In Proofgold the corresponding term root is f61607... and proposition id is b3bbd0...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__20__5

L7969

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__20__5 TMUY5RTGyX3sZPo9pFK7rSa4VY51kTP6vm5 bounty of about 25 bars ***)

L7970

Variable x : set

L7971

Variable y : set

L7972

Variable z : set

L7973

Variable w : set

L7974

Variable u : set

L7975

Variable v : set

L7976

Variable x2 : set

L7977

Variable y2 : set

L7978

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

L7979

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

L7980

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

L7981

Hypothesis H3 : SNo x

L7982

Hypothesis H4 : SNo y

L7983

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L7984

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L7985

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L7986

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L7987

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L7988

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L7989

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)))

L7990

Hypothesis H13 : SNo (g x y)

L7991

Hypothesis H14 : SNo (g (g x y) z)

L7992

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L7993

Hypothesis H16 : SNo v

L7994

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L7995

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L7996

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L7997

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L7998

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L7999

Hypothesis H22 : SNo u

L8000

Hypothesis H23 : SNo x2

L8001

Hypothesis H24 : SNo y2

L8002

Hypothesis H25 : SNo (g u (g y z))

L8003

Hypothesis H26 : SNo (g x v)

L8004

Hypothesis H27 : SNo (g x x2)

L8005

Theorem. (Conj_mul_SNo_assoc_lem1__20__5)

SNo (g u v) → w < g (g x y) z

In Proofgold the corresponding term root is 799b8f... and proposition id is fa3ddc...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__20__23

L8011

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__20__23 TMc5P4U3d4PGbJGEDfdS6UC3svi4KFgbvsS bounty of about 25 bars ***)

L8012

Variable x : set

L8013

Variable y : set

L8014

Variable z : set

L8015

Variable w : set

L8016

Variable u : set

L8017

Variable v : set

L8018

Variable x2 : set

L8019

Variable y2 : set

L8020

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

L8021

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

L8022

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

L8023

Hypothesis H3 : SNo x

L8024

Hypothesis H4 : SNo y

L8025

Hypothesis H5 : SNo z

L8026

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L8027

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L8028

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L8029

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L8030

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L8031

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L8032

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)))

L8033

Hypothesis H13 : SNo (g x y)

L8034

Hypothesis H14 : SNo (g (g x y) z)

L8035

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L8036

Hypothesis H16 : SNo v

L8037

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L8038

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L8039

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L8040

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L8041

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L8042

Hypothesis H22 : SNo u

L8043

Hypothesis H24 : SNo y2

L8044

Hypothesis H25 : SNo (g u (g y z))

L8045

Hypothesis H26 : SNo (g x v)

L8046

Hypothesis H27 : SNo (g x x2)

L8047

Theorem. (Conj_mul_SNo_assoc_lem1__20__23)

SNo (g u v) → w < g (g x y) z

In Proofgold the corresponding term root is 1cf760... and proposition id is dbb1da...

Proof:
Proof not loaded.

Beginning of Section Conj_mul_SNo_assoc_lem1__22__14

L8053

Variable g : (set → (set → set))

(*** Conj_mul_SNo_assoc_lem1__22__14 TMRXTNTyEoHqgUsZCWbvERiFkvDA8pfrZ5S bounty of about 25 bars ***)

L8054

Variable x : set

L8055

Variable y : set

L8056

Variable z : set

L8057

Variable w : set

L8058

Variable u : set

L8059

Variable v : set

L8060

Variable x2 : set

L8061

Variable y2 : set

L8062

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

L8063

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

L8064

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

L8065

Hypothesis H3 : SNo x

L8066

Hypothesis H4 : SNo y

L8067

Hypothesis H5 : SNo z

L8068

Hypothesis H6 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → g z2 (g y z) = g (g z2 y) z)

L8069

Hypothesis H7 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → g x (g z2 z) = g (g x z2) z)

L8070

Hypothesis H8 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → g x (g y z2) = g (g x y) z2)

L8071

Hypothesis H9 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev y) → g z2 (g w2 z) = g (g z2 w2) z))

L8072

Hypothesis H10 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g z2 (g y w2) = g (g z2 y) w2))

L8073

Hypothesis H11 : (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → g x (g z2 w2) = g (g x z2) w2))

L8074

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)))

L8075

Hypothesis H13 : SNo (g x y)

L8076

Hypothesis H15 : u ∈ SNoS_ (SNoLev x)

L8077

Hypothesis H16 : SNo v

L8078

Hypothesis H17 : x2 ∈ SNoS_ (SNoLev y)

L8079

Hypothesis H18 : y2 ∈ SNoS_ (SNoLev z)

L8080

Hypothesis H19 : w = g u (g y z) + g x v + - (g u v)

L8081

Hypothesis H20 : (g u (g x2 z + g y y2) + g x (v + g x2 y2)) ≤ g x (g x2 z + g y y2) + g u (v + g x2 y2)

L8082

Hypothesis H21 : (g (g u y + g x x2) z + g (g x y + g u x2) y2) < g (g x y + g u x2) z + g (g u y + g x x2) y2

L8083

Hypothesis H22 : SNo u

L8084

Hypothesis H23 : SNo x2

L8085

Hypothesis H24 : SNo y2

L8086

Hypothesis H25 : SNo (g u (g y z))

L8087

Theorem. (Conj_mul_SNo_assoc_lem1__22__14)

SNo (g x v) → w < g (g x y) z

In Proofgold the corresponding term root is 4432ed... and proposition id is 08f4fe...

Proof:
Proof not loaded.