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Proofgold Proof

pf
Let x0 of type ο be given.
Assume H0: ∀ x1 . (∃ x2 : ι → ι . MetaCat_terminal_p Permutation UnaryFuncHom struct_id struct_comp x1 x2) ⟶ x0.
Apply H0 with pack_u 1 (λ x1 . x1).
Let x1 of type ο be given.
Assume H1: ∀ x2 : ι → ι . MetaCat_terminal_p Permutation UnaryFuncHom struct_id struct_comp (pack_u 1 (λ x3 . x3)) x2 ⟶ x1.
Apply H1 with λ x2 . lam (ap x2 0) (λ x3 . 0).
Claim L2: ∀ x2 . Permutation x2 ⟶ struct_u x2
Let x2 of type ι be given.
Assume H2: Permutation x2.
Apply H2 with struct_u x2.
Assume H3: struct_u x2.
Assume H4: unpack_u_o x2 (λ x3 . bij x3 x3).
The subproof is completed by applying H3.
Claim L3: Permutation (pack_u 1 (λ x2 . x2))
Apply unknownprop_ec657b7f97f95410adb1c5a290530d603e515202daab84a65beca23cc201c12b with 1, λ x2 . x2 leaving 3 subgoals.
Let x2 of type ι be given.
Assume H3: x2 ∈ 1.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H3: x2 ∈ 1.
Let x3 of type ι be given.
Assume H4: x3 ∈ 1.
Assume H5: (λ x4 . x4) x2 = (λ x4 . x4) x3.
Apply cases_1 with x2, λ x4 . x4 = x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply cases_1 with x3, λ x4 . 0 = x4 leaving 2 subgoals.
The subproof is completed by applying H4.
Let x4 of type ι → ι → ο be given.
Assume H6: x4 0 0.
The subproof is completed by applying H6.
Let x2 of type ι be given.
Assume H3: x2 ∈ 1.
Let x3 of type ο be given.
Assume H4: ∀ x4 . and (x4 ∈ 1) (x4 = x2) ⟶ x3.
Apply H4 with x2.
Apply andI with x2 ∈ 1, x2 = x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x4 of type ι → ι → ο be given.
Assume H5: x4 x2 x2.
The subproof is completed by applying H5.
Apply unknownprop_4c2c2f02b7bba52d2c2d534bff462776dfb77dde8f7d02ce4d576ba29cf94915 with Permutation leaving 2 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L3.
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