Let x0 of type ι → (ι → ι) → (ι → ι) → ο be given.

Let x1 of type ι be given.

Let x2 of type ι → ι be given.

Let x3 of type ι → ι be given.

Assume H0: ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x3 x6 = x5 x6) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3.

Apply unknownprop_543686e66106ef7fd22156a4009f0d2a9db2343f2534c11212b55b0c2e92283e with x1, x2, x3, λ x4 x5 . x0 x4 (f482f.. (f482f.. (5fdf5.. x1 x2 x3) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (5fdf5.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3.

Apply H0 with f482f.. (f482f.. (5fdf5.. x1 x2 x3) (4ae4a.. 4a7ef..)), f482f.. (f482f.. (5fdf5.. x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.

The subproof is completed by applying unknownprop_cbe3477448d585dc62b5597e422b06d1afa24a5af2e7522929e0b9389e0afec8 with x1, x2, x3.

The subproof is completed by applying unknownprop_952db7193be9ed6add7b4388af9a31411f1f39400c1e751d83d30221f79965d8 with x1, x2, x3.

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Proofgold Proof