Definition. We define struct_b_b_r_e_e_ordered_field to be λR ⇒ struct_b_b_r_e_e R ∧ unpack_b_b_r_e_e_o R (λR plus mult leq zero one ⇒ explicit_OrderedField R zero one plus mult leq) of type set → prop.
In Proofgold the corresponding term root is 98aa98... and object id is 1de6a1...
Theorem. (MetaCat_struct_b_b_r_e_e_ordered_field)
MetaCat struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp
In Proofgold the corresponding term root is cdd98e... and proposition id is 707cbd...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_b_r_e_e_ordered_field X → struct_b_b_r_e_e X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_b_b_r_e_e_gen struct_b_b_r_e_e_ordered_field L1.
∎
Theorem. (MetaCat_struct_b_b_r_e_e_ordered_field_Forgetful)
MetaFunctor struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g)) (λX ⇒ X 0) (λX Y f ⇒ f)
In Proofgold the corresponding term root is 3151a0... and proposition id is e3e2f3...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_b_r_e_e_ordered_field X → struct_b_b_r_e_e X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_b_b_r_e_e_Forgetful_gen struct_b_b_r_e_e_ordered_field L1.
∎
Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is f4c514... and proposition id is 38cd2f...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is bd133b... and proposition id is e5992d...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 4f754f... and proposition id is d2d4c3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 1b5210... and proposition id is 79bf31...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_coequalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, coequalizer_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is b2851a... and proposition id is 9d5c1a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_equalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, equalizer_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is f0fafa... and proposition id is 7479e8...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_pushout_constr)
∃po : set → set → set → set → set → set, ∃i0 : set → set → set → set → set → set, ∃i1 : set → set → set → set → set → set, ∃copair : set → set → set → set → set → set → set → set → set, pushout_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is a00d98... and proposition id is 03b598...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_pullback_constr)
∃pb : set → set → set → set → set → set, ∃pi0 : set → set → set → set → set → set, ∃pi1 : set → set → set → set → set → set, ∃pair : set → set → set → set → set → set → set → set → set, pullback_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is f9f68c... and proposition id is f53f5e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_product_exponent)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, ∃exp : set → set → set, ∃a : set → set → set, ∃lm : set → set → set → set → set, product_exponent_constr_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 6e43cd... and proposition id is 6fb85e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_subobject_classifier)
∃one : set, ∃uniqa : set → set, ∃Omega : set, ∃tru : set, ∃ch : set → set → set → set, ∃constr : set → set → set → set → set → set → set, subobject_classifier_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is d1a11f... and proposition id is 098413...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 547494... and proposition id is a0412e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_r_e_e_ordered_field_left_adjoint_forgetful)
∃F0 : set → set, ∃F1 : set → set → set → set, ∃eta eps : set → set, MetaAdjunction_strict (λ_ ⇒ True) SetHom (λX ⇒ (lam_id X)) (λX Y Z f g ⇒ (lam_comp X f g)) struct_b_b_r_e_e_ordered_field Hom_struct_b_b_r_e_e struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is a63581... and proposition id is 7c52e6...
Proof:
The rest of the proof is missing.