Let x0 of type ι be given.

Let x1 of type ι → ι be given.

Let x2 of type ι be given.

Assume H0: prim1 x2 (a842e.. x0 (λ x3 . x1 x3)).

Apply UnionE_impred with 94f9e.. x0 (λ x3 . x1 x3), x2, ∃ x3 . and (prim1 x3 x0) (prim1 x2 (x1 x3)) leaving 2 subgoals.

The subproof is completed by applying H0.

Let x3 of type ι be given.

Assume H1: prim1 x2 x3.

Assume H2: prim1 x3 (94f9e.. x0 (λ x4 . x1 x4)).

Apply unknownprop_d908b89102f7b662c739e5a844f67efc8ae1cd05a2e9ce1e3546fa3885f40100 with x0, x1, x3, ∃ x4 . and (prim1 x4 x0) (prim1 x2 (x1 x4)) leaving 2 subgoals.

The subproof is completed by applying H2.

Let x4 of type ι be given.

Assume H3: prim1 x4 x0.

Assume H4: x3 = x1 x4.

Let x5 of type ο be given.

Assume H5: ∀ x6 . and (prim1 x6 x0) (prim1 x2 (x1 x6)) ⟶ x5.

Apply H5 with x4.

Apply andI with prim1 x4 x0, prim1 x2 (x1 x4) leaving 2 subgoals.

The subproof is completed by applying H3.

Apply H4 with λ x6 x7 . prim1 x2 x6.

The subproof is completed by applying H1.

■

Proofgold Proof