Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Let x4 of type ι → ι be given.

Let x5 of type ι → ι be given.

Let x6 of type ι be given.

Let x7 of type ι be given.

Let x8 of type ι be given.

Let x9 of type ι be given.

Assume H0: 9b6e8.. x0 x2 x4 x6 x8 = 9b6e8.. x1 x3 x5 x7 x9.

Claim L1: x1 = f482f.. (9b6e8.. x0 x2 x4 x6 x8) 4a7ef..

Apply unknownprop_be86c1fee4e42f40a4d9e9fb9d212ee7b7147d4f8c46bbc842df17b16f8cf517 with 9b6e8.. x0 x2 x4 x6 x8, x1, x3, x5, x7, x9.

The subproof is completed by applying H0.

Claim L2: x0 = x1

Apply L1 with λ x10 x11 . x0 = x11.

The subproof is completed by applying unknownprop_0663073412d695176e3e23ffbbbb5da2857252d5c0f47051c9f0c33ecb11f4b6 with x0, x2, x4, x6, x8.

Apply and5I with x0 = x1, ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11, ∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10, x6 = x7, x8 = x9 leaving 5 subgoals.

The subproof is completed by applying L2.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Let x11 of type ι be given.

Assume H4: prim1 x11 x0.

Apply unknownprop_a0d932da53985136bc4d111609d0ae5a2ed2e4f1789b9bd8811552fd687d1136 with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 . x13 = x3 x10 x11 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x10 x1

Apply L2 with λ x12 x13 . prim1 x10 x12.

The subproof is completed by applying H3.

Claim L6: prim1 x11 x1

Apply L2 with λ x12 x13 . prim1 x11 x12.

The subproof is completed by applying H4.

Apply H0 with λ x12 x13 . e3162.. (f482f.. x13 (4ae4a.. 4a7ef..)) x10 x11 = x3 x10 x11.

Let x12 of type ι → ι → ο be given.

Apply unknownprop_a0d932da53985136bc4d111609d0ae5a2ed2e4f1789b9bd8811552fd687d1136 with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Apply unknownprop_c64bca701150bd7d152f116af6a1108b34c2b6e6d2765f2dada41711a2cb9dfa with x0, x2, x4, x6, x8, x10, λ x11 x12 . x12 = x5 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x10 x1

Apply L2 with λ x11 x12 . prim1 x10 x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . f482f.. (f482f.. x12 (4ae4a.. (4ae4a.. 4a7ef..))) x10 = x5 x10.

Let x11 of type ι → ι → ο be given.

Apply unknownprop_c64bca701150bd7d152f116af6a1108b34c2b6e6d2765f2dada41711a2cb9dfa with x1, x3, x5, x7, x9, x10, λ x12 x13 . x11 x13 x12.

The subproof is completed by applying L4.

Apply unknownprop_a5be19faa2e80b8da858086292146d3f609318d0df557aec5c9ce5a6ab6b4ab0 with x0, x2, x4, x6, x8, λ x10 x11 . x11 = x7.

Apply H0 with λ x10 x11 . f482f.. x11 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) = x7.

Let x10 of type ι → ι → ο be given.

The subproof is completed by applying unknownprop_a5be19faa2e80b8da858086292146d3f609318d0df557aec5c9ce5a6ab6b4ab0 with x1, x3, x5, x7, x9, λ x11 x12 . x10 x12 x11.

Apply unknownprop_c1003d86b72ccf1c88f05bbb3160c1287244e653b755f6f0f8f6e6dea303f1e4 with x0, x2, x4, x6, x8, λ x10 x11 . x11 = x9.

Apply H0 with λ x10 x11 . f482f.. x11 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x9.

Let x10 of type ι → ι → ο be given.

The subproof is completed by applying unknownprop_c1003d86b72ccf1c88f05bbb3160c1287244e653b755f6f0f8f6e6dea303f1e4 with x1, x3, x5, x7, x9, λ x11 x12 . x10 x12 x11.

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Proofgold Proof