Definition. We define struct_b_quasigroup to be λX ⇒ struct_b X ∧ unpack_b_o X (λX' op ⇒ (∀a ∈ X', bij X' X' (λx ⇒ op a x)) ∧ (∀a ∈ X', bij X' X' (λx ⇒ op x a))) of type set → prop.
In Proofgold the corresponding term root is d858cc... and object id is fe6453...
Theorem. (MetaCat_struct_b_quasigroup)
MetaCat struct_b_quasigroup Hom_struct_b struct_id struct_comp
In Proofgold the corresponding term root is 9c0f3e... and proposition id is 58eb86...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_quasigroup X → struct_b 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_gen struct_b_quasigroup L1.
∎
Theorem. (MetaCat_struct_b_quasigroup_Forgetful)
MetaFunctor struct_b_quasigroup Hom_struct_b 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 76e37c... and proposition id is 942b10...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_quasigroup X → struct_b 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_Forgetful_gen struct_b_quasigroup L1.
∎
Proposition. (MetaCat_struct_b_quasigroup_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_b_quasigroup Hom_struct_b struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is b75197... and proposition id is 3ab710...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_b_quasigroup Hom_struct_b struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is e8ac3e... and proposition id is 19aaf2...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_b_quasigroup Hom_struct_b struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 4f5706... and proposition id is a51804...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_b_quasigroup Hom_struct_b struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is f456a6... and proposition id is dd5dca...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 2fc5db... and proposition id is 5637bb...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 422d84... and proposition id is b86357...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 25673b... and proposition id is e9148b...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is 733a7a... and proposition id is d652b9...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 6439ba... and proposition id is 58d583...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is 1e48d1... and proposition id is a4869e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_b_quasigroup Hom_struct_b struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 62eb92... and proposition id is 6a5cf8...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_quasigroup_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_quasigroup Hom_struct_b struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 805666... and proposition id is 63ebf3...
Proof:
The rest of the proof is missing.