Let x0 of type ο be given.

Assume H0: ∀ x1 : ι → ι → ι . (∃ x2 x3 : ι → ι → ι . ∃ x4 : ι → ι → ι → ι → ι → ι . MetaCat_coproduct_constr_p struct_r BinRelnHom struct_id struct_comp x1 x2 x3 x4) ⟶ x0.

Apply H0 with 3fa3a...

Let x1 of type ο be given.

Assume H1: ∀ x2 : ι → ι → ι . (∃ x3 : ι → ι → ι . ∃ x4 : ι → ι → ι → ι → ι → ι . MetaCat_coproduct_constr_p struct_r BinRelnHom struct_id struct_comp 3fa3a.. x2 x3 x4) ⟶ x1.

Apply H1 with λ x2 x3 . lam (ap x2 0) (λ x4 . Inj0 x4).

Let x2 of type ο be given.

Assume H2: ∀ x3 : ι → ι → ι . (∃ x4 : ι → ι → ι → ι → ι → ι . MetaCat_coproduct_constr_p struct_r BinRelnHom struct_id struct_comp 3fa3a.. (λ x5 x6 . lam (ap x5 0) (λ x7 . Inj0 x7)) x3 x4) ⟶ x2.

Apply H2 with λ x3 x4 . lam (ap x4 0) (λ x5 . Inj1 x5).

Let x3 of type ο be given.

Assume H3: ∀ x4 : ι → ι → ι → ι → ι → ι . MetaCat_coproduct_constr_p struct_r BinRelnHom struct_id struct_comp 3fa3a.. (λ x5 x6 . lam (ap x5 0) (λ x7 . Inj0 x7)) (λ x5 x6 . lam (ap x6 0) (λ x7 . Inj1 x7)) x4 ⟶ x3.

Apply H3 with λ x4 x5 x6 x7 x8 . lam (setsum (ap x4 0) (ap x5 0)) (λ x9 . combine_funcs (ap x4 0) (ap x5 0) (λ x10 . ap x7 x10) (λ x10 . ap x8 x10) x9).

Apply unknownprop_8014f2189a8e9a90722a83ab5f5b4d52ecd6d5c686aac8aa2eb5343a4f9f7780 with struct_r leaving 2 subgoals.

Let x4 of type ι be given.

Assume H4: struct_r x4.

The subproof is completed by applying H4.

The subproof is completed by applying unknownprop_a69d20d3ecb56be7161eda7f44e43ecc2760c2c8bbc1729b0822a57a8b1a28bb.

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Proofgold Proof