Proofgold Address

Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition explicit_Group^{explicit_Group} := λ x0 . λ x1 : ι → ι → ι . and (and (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 (x1 x3 x4) = x1 (x1 x2 x3) x4)) (∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (and (∀ x4 . x4 ∈ x0 ⟶ and (x1 x3 x4 = x4) (x1 x4 x3 = x4)) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (and (x1 x4 x6 = x3) (x1 x6 x4 = x3)) ⟶ x5) ⟶ x5)) ⟶ x2) ⟶ x2)
Theorem explicit_Group_identity_unique^{explicit_Group_identity_unique} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x1 x2 x4 = x4) ⟶ (∀ x4 . x4 ∈ x0 ⟶ x1 x4 x3 = x4) ⟶ x2 = x3 (proof)
Definition explicit_Group_identity^{explicit_Group_identity} := λ x0 . λ x1 : ι → ι → ι . prim0 (λ x2 . and (x2 ∈ x0) (and (∀ x3 . x3 ∈ x0 ⟶ and (x1 x2 x3 = x3) (x1 x3 x2 = x3)) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (and (x1 x3 x5 = x2) (x1 x5 x3 = x2)) ⟶ x4) ⟶ x4)))
Definition explicit_Group_inverse^{explicit_Group_inverse} := λ x0 . λ x1 : ι → ι → ι . λ x2 . prim0 (λ x3 . and (x3 ∈ x0) (and (x1 x2 x3 = explicit_Group_identity x0 x1) (x1 x3 x2 = explicit_Group_identity x0 x1)))
Known Eps_i_ex^Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Theorem explicit_Group_identity_prop^{explicit_Group_identity_prop} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ and (explicit_Group_identity x0 x1 ∈ x0) (and (∀ x2 . x2 ∈ x0 ⟶ and (x1 (explicit_Group_identity x0 x1) x2 = x2) (x1 x2 (explicit_Group_identity x0 x1) = x2)) (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (and (x1 x2 x4 = explicit_Group_identity x0 x1) (x1 x4 x2 = explicit_Group_identity x0 x1)) ⟶ x3) ⟶ x3)) (proof)
Theorem explicit_Group_identity_in^{explicit_Group_identity_in} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ explicit_Group_identity x0 x1 ∈ x0 (proof)
Theorem explicit_Group_identity_lid^{explicit_Group_identity_lid} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 (explicit_Group_identity x0 x1) x2 = x2 (proof)
Theorem explicit_Group_identity_rid^{explicit_Group_identity_rid} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 x2 (explicit_Group_identity x0 x1) = x2 (proof)
Theorem explicit_Group_identity_invex^{explicit_Group_identity_invex} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (and (x1 x2 x4 = explicit_Group_identity x0 x1) (x1 x4 x2 = explicit_Group_identity x0 x1)) ⟶ x3) ⟶ x3 (proof)
Theorem explicit_Group_inverse_prop^{explicit_Group_inverse_prop} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ and (explicit_Group_inverse x0 x1 x2 ∈ x0) (and (x1 x2 (explicit_Group_inverse x0 x1 x2) = explicit_Group_identity x0 x1) (x1 (explicit_Group_inverse x0 x1 x2) x2 = explicit_Group_identity x0 x1)) (proof)
Theorem explicit_Group_inverse_in^{explicit_Group_inverse_in} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ explicit_Group_inverse x0 x1 x2 ∈ x0 (proof)
Theorem explicit_Group_inverse_rinv^{explicit_Group_inverse_rinv} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 x2 (explicit_Group_inverse x0 x1 x2) = explicit_Group_identity x0 x1 (proof)
Theorem explicit_Group_inverse_linv^{explicit_Group_inverse_linv} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 (explicit_Group_inverse x0 x1 x2) x2 = explicit_Group_identity x0 x1 (proof)
Theorem explicit_Group_lcancel^{explicit_Group_lcancel} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 = x1 x2 x4 ⟶ x3 = x4 (proof)
Theorem explicit_Group_rcancel^{explicit_Group_rcancel} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x4 = x1 x3 x4 ⟶ x2 = x3 (proof)
Theorem explicit_Group_rinv_rev^{explicit_Group_rinv_rev} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 = explicit_Group_identity x0 x1 ⟶ x3 = explicit_Group_inverse x0 x1 x2 (proof)
Theorem explicit_Group_inv_com^{explicit_Group_inv_com} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 = explicit_Group_identity x0 x1 ⟶ x1 x3 x2 = explicit_Group_identity x0 x1 (proof)
Theorem explicit_Group_inv_rev2^{explicit_Group_inv_rev2} : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 (x1 x2 x3) (x1 x2 x3) = explicit_Group_identity x0 x1 ⟶ x1 (x1 x3 x2) (x1 x3 x2) = explicit_Group_identity x0 x1 (proof)
Definition explicit_abelian^{explicit_abelian} := λ x0 . λ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 = x1 x3 x2
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem explicit_Group_repindep_imp^{explicit_Group_repindep_imp} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ explicit_Group x0 x1 ⟶ explicit_Group x0 x2 (proof)
Theorem explicit_Group_identity_repindep^{explicit_Group_identity_repindep} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ explicit_Group x0 x1 ⟶ explicit_Group_identity x0 x1 = explicit_Group_identity x0 x2 (proof)
Theorem explicit_Group_inverse_repindep^{explicit_Group_inverse_repindep} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ explicit_Group x0 x1 ⟶ ∀ x3 . x3 ∈ x0 ⟶ explicit_Group_inverse x0 x1 x3 = explicit_Group_inverse x0 x2 x3 (proof)
Theorem explicit_abelian_repindep_imp^{explicit_abelian_repindep_imp} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ explicit_abelian x0 x1 ⟶ explicit_abelian x0 x2 (proof)
Param iff^iff : ο → ο → ο
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem explicit_Group_repindep^{explicit_Group_repindep} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ iff (explicit_Group x0 x1) (explicit_Group x0 x2) (proof)
Theorem explicit_abelian_repindep^{explicit_abelian_repindep} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ iff (explicit_abelian x0 x1) (explicit_abelian x0 x2) (proof)
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 unpack_b_o^unpack_b_o : ι → (ι → (ι → ι → ι) → ο) → ο
Definition Group^Group := λ x0 . and (struct_b x0) (unpack_b_o x0 explicit_Group)
Definition abelian_Group^{abelian_Group} := λ x0 . and (Group x0) (unpack_b_o x0 explicit_abelian)
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^prop_ext : ∀ x0 x1 : ο . iff x0 x1 ⟶ x0 = x1
Theorem Group_unpack_eq^{Group_unpack_eq} : ∀ x0 . ∀ x1 : ι → ι → ι . unpack_b_o (pack_b x0 x1) explicit_Group = explicit_Group x0 x1 (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 GroupI^GroupI : ∀ x0 . ∀ x1 : ι → ι → ι . explicit_Group x0 x1 ⟶ Group (pack_b x0 x1) (proof)
Theorem GroupE^GroupE : ∀ x0 . ∀ x1 : ι → ι → ι . Group (pack_b x0 x1) ⟶ explicit_Group x0 x1 (proof)
Theorem abelian_Group_unpack_eq^{abelian_Group_unpack_eq} : ∀ x0 . ∀ x1 : ι → ι → ι . unpack_b_o (pack_b x0 x1) explicit_abelian = explicit_abelian x0 x1 (proof)
Theorem abelian_Group_E^{abelian_Group_E} : ∀ x0 . ∀ x1 : ι → ι → ι . abelian_Group (pack_b x0 x1) ⟶ and (Group (pack_b x0 x1)) (explicit_abelian x0 x1) (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition explicit_subgroup^{explicit_subgroup} := λ x0 . λ x1 : ι → ι → ι . λ x2 . and (Group (pack_b x2 x1)) (x2 ⊆ x0)
Definition explicit_normal^{explicit_normal} := λ x0 . λ x1 : ι → ι → ι . λ x2 . ∀ x3 . x3 ∈ x0 ⟶ {x1 x3 (x1 x4 (explicit_Group_inverse x0 x1 x3))|x4 ∈ x2} ⊆ x2
Theorem explicit_subgroup_test^{explicit_subgroup_test} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . Group (pack_b x0 x1) ⟶ x2 ⊆ x0 ⟶ explicit_Group_identity x0 x1 ∈ x2 ⟶ (∀ x3 . x3 ∈ x2 ⟶ explicit_Group_inverse x0 x1 x3 ∈ x2) ⟶ (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ x1 x3 x4 ∈ x2) ⟶ explicit_subgroup x0 x1 x2 (proof)
Theorem explicit_subgroup_identity_eq^{explicit_subgroup_identity_eq} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . Group (pack_b x0 x1) ⟶ explicit_subgroup x0 x1 x2 ⟶ explicit_Group_identity x0 x1 = explicit_Group_identity x2 x1 (proof)
Theorem explicit_subgroup_inv_eq^{explicit_subgroup_inv_eq} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . Group (pack_b x0 x1) ⟶ explicit_subgroup x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ explicit_Group_inverse x0 x1 x3 = explicit_Group_inverse x2 x1 x3 (proof)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Theorem explicit_abelian_normal^{explicit_abelian_normal} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . Group (pack_b x0 x1) ⟶ explicit_subgroup x0 x1 x2 ⟶ explicit_abelian x0 x1 ⟶ explicit_normal x0 x1 x2 (proof)
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem explicit_normal_repindep_imp^{explicit_normal_repindep_imp} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Group x1 x2 ⟶ x0 ⊆ x1 ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ explicit_normal x1 x2 x0 ⟶ explicit_normal x1 x3 x0 (proof)
Definition subgroup^subgroup := λ x0 x1 . and (and (struct_b x1) (struct_b x0)) (unpack_b_o x1 (λ x2 . λ x3 : ι → ι → ι . unpack_b_o x0 (λ x4 . λ x5 : ι → ι → ι . and (and (x0 = pack_b x4 x3) (Group (pack_b x4 x3))) (x4 ⊆ x2))))
Param unpack_b_i^unpack_b_i : ι → (ι → CT2 ι) → ι
Param Sep^Sep : ι → (ι → ο) → ι
Param omega^omega : ι
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param ordsucc^ordsucc : ι → ι
Param ap^ap : ι → ι → ι
Definition subgroup_index^{subgroup_index} := λ x0 x1 . unpack_b_i x1 (λ x2 . λ x3 : ι → ι → ι . {x4 ∈ omega|∀ x5 : ο . (∀ x6 . and (x6 ∈ setexp x2 (ordsucc x4)) (∀ x7 . x7 ∈ ordsucc x4 ⟶ ∀ x8 . x8 ∈ ordsucc x4 ⟶ (x7 = x8 ⟶ ∀ x9 : ο . x9) ⟶ ∀ x9 . x9 ∈ ap x0 0 ⟶ ∀ x10 . x10 ∈ ap x0 0 ⟶ x3 (ap x6 x7) x9 = x3 (ap x6 x8) x10 ⟶ ∀ x11 : ο . x11) ⟶ x5) ⟶ x5})
Definition normal_subgroup^{normal_subgroup} := λ x0 x1 . and (subgroup x0 x1) (unpack_b_o x1 (λ x2 . λ x3 : ι → ι → ι . unpack_b_o x0 (λ x4 . λ x5 : ι → ι → ι . explicit_normal x2 x3 x4)))
Theorem 206a1.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . unpack_b_o (pack_b x0 x2) (λ x5 . λ x6 : ι → ι → ι . and (and (pack_b x0 x2 = pack_b x5 x3) (Group (pack_b x5 x3))) (x5 ⊆ x1)) = and (and (pack_b x0 x2 = pack_b x0 x3) (Group (pack_b x0 x3))) (x0 ⊆ x1) (proof)
Known prop_ext_2^prop_ext_2 : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ x0 = x1
Known pack_b_ext^pack_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ pack_b x0 x1 = pack_b x0 x2
Theorem 39d2e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . unpack_b_o (pack_b x1 x3) (λ x5 . λ x6 : ι → ι → ι . unpack_b_o (pack_b x0 x2) (λ x7 . λ x8 : ι → ι → ι . and (and (pack_b x0 x2 = pack_b x7 x6) (Group (pack_b x7 x6))) (x7 ⊆ x5))) = and (and (pack_b x0 x2 = pack_b x0 x3) (Group (pack_b x0 x3))) (x0 ⊆ x1) (proof)
Theorem pack_b_subgroup_E^{pack_b_subgroup_E} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . subgroup (pack_b x0 x2) (pack_b x1 x3) ⟶ and (pack_b x0 x2 = pack_b x0 x3) (explicit_subgroup x1 x3 x0) (proof)
Theorem subgroup_E^subgroup_E : ∀ x0 x1 . subgroup x0 x1 ⟶ ∀ x2 : ι → ι → ο . (∀ x3 x4 . ∀ x5 : ι → ι → ι . (∀ x6 . x6 ∈ x4 ⟶ ∀ x7 . x7 ∈ x4 ⟶ x5 x6 x7 ∈ x4) ⟶ Group (pack_b x3 x5) ⟶ x3 ⊆ x4 ⟶ x2 (pack_b x3 x5) (pack_b x4 x5)) ⟶ x2 x0 x1 (proof)
Theorem 884ec.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Group x1 x2 ⟶ x0 ⊆ x1 ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ explicit_normal x1 x2 x0 = explicit_normal x1 x3 x0 (proof)
Theorem 4d7f2.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . unpack_b_o (pack_b x0 x2) (λ x5 . λ x6 : ι → ι → ι . explicit_normal x1 x3 x5) = explicit_normal x1 x3 x0 (proof)
Theorem 84827.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . Group (pack_b x1 x3) ⟶ x0 ⊆ x1 ⟶ unpack_b_o (pack_b x1 x3) (λ x5 . λ x6 : ι → ι → ι . unpack_b_o (pack_b x0 x2) (λ x7 . λ x8 : ι → ι → ι . explicit_normal x5 x6 x7)) = explicit_normal x1 x3 x0 (proof)
Theorem abelian_group_normal_subgroup^{abelian_group_normal_subgroup} : ∀ x0 . abelian_Group x0 ⟶ ∀ x1 . subgroup x1 x0 ⟶ normal_subgroup x1 x0 (proof)
Known pack_b_inj^pack_b_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . pack_b x0 x2 = pack_b x1 x3 ⟶ and (x0 = x1) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = x3 x4 x5)
Theorem subgroup_transitive^{subgroup_transitive} : ∀ x0 x1 x2 . subgroup x0 x1 ⟶ subgroup x1 x2 ⟶ subgroup x0 x2 (proof)
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)
Param lam^Sigma : ι → (ι → ι) → ι
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Theorem 6a414.. : ∀ x0 x1 . x1 ∈ {x2 ∈ setexp x0 x0|bij x0 x0 (ap x2)} ⟶ ∀ x2 . x2 ∈ {x3 ∈ setexp x0 x0|bij x0 x0 (ap x3)} ⟶ lam x0 (λ x3 . ap x2 (ap x1 x3)) ∈ {x3 ∈ setexp x0 x0|bij x0 x0 (ap x3)} (proof)
Theorem 86033.. : ∀ x0 . lam x0 (λ x1 . x1) ∈ {x1 ∈ setexp x0 x0|bij x0 x0 (ap x1)} (proof)
Param inv^inv : ι → (ι → ι) → ι → ι
Known encode_u_ext^encode_u_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ lam x0 x1 = lam x0 x2
Known inj_linv_coddep : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ ∀ x3 . x3 ∈ x0 ⟶ inv x0 x2 (x2 x3) = x3
Known surj_rinv^surj_rinv : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ ∀ x3 . x3 ∈ x1 ⟶ and (inv x0 x2 x3 ∈ x0) (x2 (inv x0 x2 x3) = x3)
Theorem 11cb6.. : ∀ x0 x1 . x1 ∈ {x2 ∈ setexp x0 x0|bij x0 x0 (ap x2)} ⟶ and (and (lam x0 (inv x0 (ap x1)) ∈ {x2 ∈ setexp x0 x0|bij x0 x0 (ap x2)}) (lam x0 (λ x3 . ap (lam x0 (inv x0 (ap x1))) (ap x1 x3)) = lam x0 (λ x3 . x3))) (lam x0 (λ x3 . ap x1 (ap (lam x0 (inv x0 (ap x1))) x3)) = lam x0 (λ x3 . x3)) (proof)
Known Pi_ext^Pi_ext : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ ∀ x3 . x3 ∈ Pi x0 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ ap x2 x4 = ap x3 x4) ⟶ x2 = x3
Theorem explicit_Group_symgroup^{explicit_Group_symgroup} : ∀ x0 . explicit_Group {x1 ∈ setexp x0 x0|bij x0 x0 (ap x1)} (λ x1 x2 . lam x0 (λ x3 . ap x2 (ap x1 x3))) (proof)
Theorem explicit_Group_symgroup_id_eq^{explicit_Group_symgroup_id_eq} : ∀ x0 . explicit_Group_identity {x2 ∈ setexp x0 x0|bij x0 x0 (ap x2)} (λ x2 x3 . lam x0 (λ x4 . ap x3 (ap x2 x4))) = lam x0 (λ x2 . x2) (proof)
Theorem explicit_Group_symgroup_inv_eq^{explicit_Group_symgroup_inv_eq} : ∀ x0 x1 . x1 ∈ {x2 ∈ setexp x0 x0|bij x0 x0 (ap x2)} ⟶ explicit_Group_inverse {x3 ∈ setexp x0 x0|bij x0 x0 (ap x3)} (λ x3 x4 . lam x0 (λ x5 . ap x4 (ap x3 x5))) x1 = lam x0 (inv x0 (ap x1)) (proof)
Theorem 801b9.. : ∀ x0 x1 . x1 ⊆ x0 ⟶ ∀ x2 . x2 ∈ {x3 ∈ setexp x0 x0|and (bij x0 x0 (ap x3)) (∀ x4 . x4 ∈ x1 ⟶ ap x3 x4 = x4)} ⟶ ∀ x3 . x3 ∈ {x4 ∈ setexp x0 x0|and (bij x0 x0 (ap x4)) (∀ x5 . x5 ∈ x1 ⟶ ap x4 x5 = x5)} ⟶ lam x0 (λ x4 . ap x3 (ap x2 x4)) ∈ {x4 ∈ setexp x0 x0|and (bij x0 x0 (ap x4)) (∀ x5 . x5 ∈ x1 ⟶ ap x4 x5 = x5)} (proof)
Theorem explicit_subgroup_symgroup_fixing^{explicit_subgroup_symgroup_fixing} : ∀ x0 x1 . x1 ⊆ x0 ⟶ explicit_subgroup {x2 ∈ setexp x0 x0|bij x0 x0 (ap x2)} (λ x2 x3 . lam x0 (λ x4 . ap x3 (ap x2 x4))) {x2 ∈ setexp x0 x0|and (bij x0 x0 (ap x2)) (∀ x3 . x3 ∈ x1 ⟶ ap x2 x3 = x3)} (proof)
Definition symgroup^symgroup := λ x0 . pack_b {x1 ∈ setexp x0 x0|bij x0 x0 (ap x1)} (λ x1 x2 . lam x0 (λ x3 . ap x2 (ap x1 x3)))
Definition symgroup_fixing^{symgroup_fixing} := λ x0 x1 . pack_b {x2 ∈ setexp x0 x0|and (bij x0 x0 (ap x2)) (∀ x3 . x3 ∈ x1 ⟶ ap x2 x3 = x3)} (λ x2 x3 . lam x0 (λ x4 . ap x3 (ap x2 x4)))
Theorem Group_symgroup^{Group_symgroup} : ∀ x0 . Group (symgroup x0) (proof)
Theorem Group_symgroup_fixing^{Group_symgroup_fixing} : ∀ x0 x1 . x1 ⊆ x0 ⟶ Group (symgroup_fixing x0 x1) (proof)
Theorem subgroup_symgroup_fixing^{subgroup_symgroup_fixing} : ∀ x0 x1 . x1 ⊆ x0 ⟶ subgroup (symgroup_fixing x0 x1) (symgroup x0) (proof)
Theorem subgroup_symgroup_fixing2^{subgroup_symgroup_fixing2} : ∀ x0 x1 x2 . x2 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ subgroup (symgroup_fixing x0 x1) (symgroup_fixing x0 x2) (proof)
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Param If_i^If_i : ο → ι → ι → ι
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known tuple_3_0_eq^tuple_3_0_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 0 = x0
Known In_2_3^In_2_3 : 2 ∈ 3
Known tuple_3_2_eq^tuple_3_2_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 2 = x2
Known In_0_3^In_0_3 : 0 ∈ 3
Known In_0_1^In_0_1 : 0 ∈ 1
Known tuple_3_in_A_3^{tuple_3_in_A_3} : ∀ x0 x1 x2 x3 . x0 ∈ x3 ⟶ x1 ∈ x3 ⟶ x2 ∈ x3 ⟶ lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)) ∈ setexp x3 3
Known In_1_3^In_1_3 : 1 ∈ 3
Known tuple_3_bij_3^{tuple_3_bij_3} : ∀ x0 x1 x2 . x0 ∈ 3 ⟶ x1 ∈ 3 ⟶ x2 ∈ 3 ⟶ (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ bij 3 3 (ap (lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) x1 x2))))
Known neq_0_2^neq_0_2 : 0 = 2 ⟶ ∀ x0 : ο . x0
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem nonnormal_subgroup^{nonnormal_subgroup} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 . and (and (Group x3) (subgroup x1 x3)) (not (normal_subgroup x1 x3)) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition normal_subgroup_equiv^{normal_subgroup_equiv} := λ x0 x1 x2 x3 . unpack_b_o x0 (λ x4 . λ x5 : ι → ι → ι . and (and (x2 ∈ x4) (x3 ∈ x4)) (x5 x2 (explicit_Group_inverse x4 x5 x3) ∈ ap x1 0))
Param quotient^quotient : (ι → ι → ο) → ι → ο
Param canonical_elt^{canonical_elt} : (ι → ι → ο) → ι → ι
Definition quotient_Group^{quotient_Group} := λ x0 x1 . unpack_b_i x0 (λ x2 . λ x3 : ι → ι → ι . pack_b (Sep x2 (quotient (normal_subgroup_equiv x0 x1))) (λ x4 x5 . canonical_elt (normal_subgroup_equiv x0 x1) (x3 x4 x5)))
Definition trivial_Group_p^{trivial_Group_p} := λ x0 . and (Group x0) (∀ x1 . x1 ∈ ap x0 0 ⟶ ∀ x2 . x2 ∈ ap x0 0 ⟶ x1 = x2)
Definition 4925b.. := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (and (and (and (∀ x5 . x5 ∈ ordsucc x2 ⟶ Group (ap x4 x5)) (∀ x5 . x5 ∈ x2 ⟶ normal_subgroup (ap x4 x5) (ap x4 (ordsucc x5)))) (∀ x5 . x5 ∈ x2 ⟶ abelian_Group (quotient_Group (ap x4 (ordsucc x5)) (ap x4 x5)))) (x0 = ap x4 x2)) (trivial_Group_p (ap x4 0)) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1