Proofgold Address

Param pack_b^pack_b : ι → CT2 ι
Definition struct_b^struct_b := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ x1 (pack_b x2 x3)) ⟶ x1 x0
Param MetaCat_initial_p^initial_p : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ι → (ι → ι) → ο
Param MagmaHom^Hom_struct_b : ι → ι → ι → ο
Param struct_id^struct_id : ι → ι
Param struct_comp^struct_comp : ι → ι → ι → ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Known 93a0d.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ x0 (pack_b 0 (λ x1 x2 . x1)) ⟶ MetaCat_initial_p x0 MagmaHom struct_id struct_comp (pack_b 0 (λ x1 x2 . x1)) (λ x1 . lam 0 (λ x2 . x2))
Param ordsucc^ordsucc : ι → ι
Param MetaCat_terminal_p^terminal_p : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ι → (ι → ι) → ο
Param ap^ap : ι → ι → ι
Known dfc93.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ x0 (pack_b 1 (λ x1 x2 . x1)) ⟶ MetaCat_terminal_p x0 MagmaHom struct_id struct_comp (pack_b 1 (λ x1 x2 . x1)) (λ x1 . lam (ap x1 0) (λ x2 . 0))
Param 3d151.. : ι → ι → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param If_i^If_i : ο → ι → ι → ι
Known fe106.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . ∀ x3 : ι → ι → ι . 3d151.. (pack_b x0 x1) (pack_b x2 x3) = pack_b (setprod x0 x2) (λ x5 x6 . lam 2 (λ x7 . If_i (x7 = 0) (x1 (ap x5 0) (ap x6 0)) (x3 (ap x5 1) (ap x6 1))))
Param MetaCat_product_constr_p^{product_constr_p} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → ο
Known 91c36.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ (∀ x1 x2 . x0 x1 ⟶ x0 x2 ⟶ x0 (3d151.. x1 x2)) ⟶ MetaCat_product_constr_p x0 MagmaHom struct_id struct_comp 3d151.. (λ x1 x2 . lam (setprod (ap x1 0) (ap x2 0)) (λ x3 . ap x3 0)) (λ x1 x2 . lam (setprod (ap x1 0) (ap x2 0)) (λ x3 . ap x3 1)) (λ x1 x2 x3 x4 x5 . lam (ap x3 0) (λ x6 . lam 2 (λ x7 . If_i (x7 = 0) (ap x4 x6) (ap x5 x6))))
Param 32592.. : ι → ι → ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Known e123b.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 x3 x4 . 32592.. (pack_b x0 x1) x2 x3 x4 = pack_b {x6 ∈ x0|ap x3 x6 = ap x4 x6} x1
Param MetaCat_equalizer_struct_p^{equalizer_constr_p} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι → ι) → ο
Known d19d6.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ (∀ x1 x2 x3 x4 . x0 x1 ⟶ x0 x2 ⟶ MagmaHom x1 x2 x3 ⟶ MagmaHom x1 x2 x4 ⟶ x0 (32592.. x1 x2 x3 x4)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p x0 MagmaHom struct_id struct_comp x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition bij^bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Known bijI^bijI : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ bij x0 x1 x2
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known In_0_1^In_0_1 : 0 ∈ 1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem fdf81.. : ∀ x0 : ι → ι . x0 0 = 0 ⟶ bij 1 1 x0 (proof)
Theorem b3818.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x3 x4) ⟶ bij x0 x1 x2 ⟶ bij x0 x1 x3 (proof)
Param unpack_b_o^unpack_b_o : ι → (ι → (ι → ι → ι) → ο) → ο
Definition Quasigroup^{struct_b_quasigroup} := λ x0 . and (struct_b x0) (unpack_b_o x0 (λ x1 . λ x2 : ι → ι → ι . and (∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (x2 x3)) (∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (λ x4 . x2 x4 x3))))
Known unpack_b_o_eq^{unpack_b_o_eq} : ∀ x0 : ι → (ι → ι → ι) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . (∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_b_o (pack_b x1 x2) x0 = x0 x1 x2
Known prop_ext_2^prop_ext_2 : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ x0 = x1
Theorem 162c3.. : ∀ x0 . ∀ x1 : ι → ι → ι . unpack_b_o (pack_b x0 x1) (λ x3 . λ x4 : ι → ι → ι . and (∀ x5 . x5 ∈ x3 ⟶ bij x3 x3 (x4 x5)) (∀ x5 . x5 ∈ x3 ⟶ bij x3 x3 (λ x6 . x4 x6 x5))) = and (∀ x3 . x3 ∈ x0 ⟶ bij x0 x0 (x1 x3)) (∀ x3 . x3 ∈ x0 ⟶ bij x0 x0 (λ x4 . x1 x4 x3)) (proof)
Known pack_struct_b_I^{pack_struct_b_I} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ struct_b (pack_b x0 x1)
Theorem 1be6f.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ (∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (x1 x2)) ⟶ (∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (λ x3 . x1 x3 x2)) ⟶ Quasigroup (pack_b x0 x1) (proof)
Theorem ed13e.. : ∀ x0 . Quasigroup x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (x3 x4)) ⟶ (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (λ x5 . x3 x5 x4)) ⟶ x1 (pack_b x2 x3)) ⟶ x1 x0 (proof)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem 3ab71..^{MetaCat_struct_b_quasigroup_initial} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_initial_p Quasigroup MagmaHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 19aaf..^{MetaCat_struct_b_quasigroup_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p Quasigroup MagmaHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known tuple_2_inj^tuple_2_inj : ∀ x0 x1 x2 x3 . lam 2 (λ x5 . If_i (x5 = 0) x0 x1) = lam 2 (λ x5 . If_i (x5 = 0) x2 x3) ⟶ and (x0 = x2) (x1 = x3)
Known tuple_Sigma_eta^{tuple_Sigma_eta} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) (ap x2 0) (ap x2 1)) = x2
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 x1
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known tuple_2_Sigma^{tuple_2_Sigma} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 x2 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ lam x0 x1
Theorem dd5dc..^{MetaCat_struct_b_quasigroup_product_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p Quasigroup MagmaHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Known 2cd8d..^{Hom_struct_b_pack} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . MagmaHom (pack_b x0 x2) (pack_b x1 x3) x4 = and (x4 ∈ setexp x1 x0) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ap x4 (x2 x6 x7) = x3 (ap x4 x6) (ap x4 x7))
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Theorem b8635..^{MetaCat_struct_b_quasigroup_equalizer_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p Quasigroup MagmaHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param MetaCat^MetaCat : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ο
Known 58eb8..^{MetaCat_struct_b_quasigroup} : MetaCat Quasigroup MagmaHom struct_id struct_comp
Param MetaCat_pullback_struct_p^{pullback_constr_p} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο
Known ed2b0..^{product_equalizer_pullback_constr_ex} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . MetaCat x0 x1 x2 x3 ⟶ (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p x0 x1 x2 x3 x5 x7 x9 ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ (∀ x4 : ο . (∀ x5 : ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι . (∀ x8 : ο . (∀ x9 : ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p x0 x1 x2 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ ∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_struct_p x0 x1 x2 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4
Theorem d652b..^{MetaCat_struct_b_quasigroup_pullback_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι → ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_struct_p Quasigroup MagmaHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition 6e587..^{struct_b_loop} := λ x0 . and (struct_b x0) (unpack_b_o x0 (λ x1 . λ x2 : ι → ι → ι . and (and (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (∀ x5 . x5 ∈ x1 ⟶ and (x2 x5 x4 = x5) (x2 x4 x5 = x5)) ⟶ x3) ⟶ x3) (∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (x2 x3))) (∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (λ x4 . x2 x4 x3))))
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem 07b60.. : ∀ x0 . ∀ x1 : ι → ι → ι . unpack_b_o (pack_b x0 x1) (λ x3 . λ x4 : ι → ι → ι . and (and (∀ x5 : ο . (∀ x6 . and (x6 ∈ x3) (∀ x7 . x7 ∈ x3 ⟶ and (x4 x7 x6 = x7) (x4 x6 x7 = x7)) ⟶ x5) ⟶ x5) (∀ x5 . x5 ∈ x3 ⟶ bij x3 x3 (x4 x5))) (∀ x5 . x5 ∈ x3 ⟶ bij x3 x3 (λ x6 . x4 x6 x5))) = and (and (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (∀ x5 . x5 ∈ x0 ⟶ and (x1 x5 x4 = x5) (x1 x4 x5 = x5)) ⟶ x3) ⟶ x3) (∀ x3 . x3 ∈ x0 ⟶ bij x0 x0 (x1 x3))) (∀ x3 . x3 ∈ x0 ⟶ bij x0 x0 (λ x4 . x1 x4 x3)) (proof)
Theorem f0660.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ (∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (∀ x4 . x4 ∈ x0 ⟶ and (x1 x4 x3 = x4) (x1 x3 x4 = x4)) ⟶ x2) ⟶ x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (x1 x2)) ⟶ (∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (λ x3 . x1 x3 x2)) ⟶ 6e587.. (pack_b x0 x1) (proof)
Theorem 1b12a.. : ∀ x0 . 6e587.. x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . ∀ x4 . x4 ∈ x2 ⟶ (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x5 x6 ∈ x2) ⟶ (∀ x5 . x5 ∈ x2 ⟶ and (x3 x5 x4 = x5) (x3 x4 x5 = x5)) ⟶ (∀ x5 . x5 ∈ x2 ⟶ bij x2 x2 (x3 x5)) ⟶ (∀ x5 . x5 ∈ x2 ⟶ bij x2 x2 (λ x6 . x3 x6 x5)) ⟶ x1 (pack_b x2 x3)) ⟶ x1 x0 (proof)
Theorem 7fbaf..^{MetaCat_struct_b_loop_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p 6e587.. MagmaHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 6133c..^{MetaCat_struct_b_loop_product_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p 6e587.. MagmaHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem f1b36..^{MetaCat_struct_b_loop_equalizer_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p 6e587.. MagmaHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known 8207e..^{MetaCat_struct_b_loop} : MetaCat 6e587.. MagmaHom struct_id struct_comp
Theorem 2614c..^{MetaCat_struct_b_loop_pullback_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι → ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_struct_p 6e587.. MagmaHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)