Let x0 of type ι be given.

Let x1 of type ι → ι be given.

Let x2 of type ι → ι be given.

Let x3 of type ι → ο be given.

Let x4 of type ι be given.

Assume H0: 57517.. (7e46e.. x0 x1 x2 x3 x4).

Apply H0 with λ x5 . x5 = 7e46e.. x0 x1 x2 x3 x4 ⟶ prim1 x4 x0 leaving 2 subgoals.

Let x5 of type ι be given.

Let x6 of type ι → ι be given.

Assume H1: ∀ x7 . prim1 x7 x5 ⟶ prim1 (x6 x7) x5.

Let x7 of type ι → ι be given.

Assume H2: ∀ x8 . prim1 x8 x5 ⟶ prim1 (x7 x8) x5.

Let x8 of type ι → ο be given.

Let x9 of type ι be given.

Assume H3: prim1 x9 x5.

Assume H4: 7e46e.. x5 x6 x7 x8 x9 = 7e46e.. x0 x1 x2 x3 x4.

Apply unknownprop_4a59ba2d6fffe13343e206ee7d3ec36edfab901a6fd6cd245f95159b1befe641 with x5, x0, x6, x1, x7, x2, x8, x3, x9, x4, prim1 x4 x0 leaving 2 subgoals.

The subproof is completed by applying H4.

Assume H5: and (and (and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10)) (∀ x10 . prim1 x10 x5 ⟶ x7 x10 = x2 x10)) (∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10).

Apply H5 with x9 = x4 ⟶ prim1 x4 x0.

Assume H6: and (and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10)) (∀ x10 . prim1 x10 x5 ⟶ x7 x10 = x2 x10).

Apply H6 with (∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10) ⟶ x9 = x4 ⟶ prim1 x4 x0.

Assume H7: and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10).

Apply H7 with (∀ x10 . prim1 x10 x5 ⟶ x7 x10 = x2 x10) ⟶ (∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10) ⟶ x9 = x4 ⟶ prim1 x4 x0.

Assume H8: x5 = x0.

Assume H9: ∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10.

Assume H10: ∀ x10 . prim1 x10 x5 ⟶ x7 x10 = x2 x10.

Assume H11: ∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10.

Assume H12: x9 = x4.

Apply H8 with λ x10 x11 . prim1 x4 x10.

Apply H12 with λ x10 x11 . prim1 x10 x5.

The subproof is completed by applying H3.

Let x5 of type ι → ι → ο be given.

Assume H1: x5 (7e46e.. x0 x1 x2 x3 x4) (7e46e.. x0 x1 x2 x3 x4).

The subproof is completed by applying H1.

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Proofgold Proof