Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ο be given.

Let x3 of type ι → ι → ο be given.

Let x4 of type ι → ι → ο be given.

Let x5 of type ι → ι → ο 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: 0d9e7.. x0 x2 x4 x6 x8 = 0d9e7.. x1 x3 x5 x7 x9.

Claim L1: x1 = f482f.. (0d9e7.. x0 x2 x4 x6 x8) 4a7ef..

Apply unknownprop_7f04e4e3e866c7d2344cfd2d751519732676a67a854e408d66db25d805444000 with 0d9e7.. 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_e30283de5e06c43feeedf7ea28f31302b2b9c54b266fa07d32b4a8647937baf2 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 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11, 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_bf3a81e508dd5bb7ebd0d922aa476a754ab01577312522215ed550d7a6c72b06 with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 : ο . 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 . 2b2e3.. (f482f.. x13 (4ae4a.. 4a7ef..)) x10 x11 = x3 x10 x11.

Let x12 of type ο → ο → ο be given.

Apply unknownprop_bf3a81e508dd5bb7ebd0d922aa476a754ab01577312522215ed550d7a6c72b06 with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 : ο . 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.

Let x11 of type ι be given.

Assume H4: prim1 x11 x0.

Apply unknownprop_d478b39a50d95c7bfa6335b02cb600d8bcf61e1565ccfaa353e4998231071c0e with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 : ο . x13 = x5 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 . 2b2e3.. (f482f.. x13 (4ae4a.. (4ae4a.. 4a7ef..))) x10 x11 = x5 x10 x11.

Let x12 of type ο → ο → ο be given.

Apply unknownprop_d478b39a50d95c7bfa6335b02cb600d8bcf61e1565ccfaa353e4998231071c0e with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 : ο . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Apply unknownprop_a74292f97535bb6becd1b270890a3d3a46394d140b4094c8913e6d9ae8b70b41 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_a74292f97535bb6becd1b270890a3d3a46394d140b4094c8913e6d9ae8b70b41 with x1, x3, x5, x7, x9, λ x11 x12 . x10 x12 x11.

Apply unknownprop_5375e18e6b66f3447ee2d8a0cd062eefda6605eebcb4137b7f34fa84355c3421 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_5375e18e6b66f3447ee2d8a0cd062eefda6605eebcb4137b7f34fa84355c3421 with x1, x3, x5, x7, x9, λ x11 x12 . x10 x12 x11.

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Proofgold Proof