Let x0 of type ι 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.

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.

Assume H0: 1bcc7.. x0 x2 x4 x6 = 1bcc7.. x1 x3 x5 x7.

Claim L1: x1 = f482f.. (1bcc7.. x0 x2 x4 x6) 4a7ef..

Apply unknownprop_693638c467dbd912f9c0a5e25f509ff870b163d0bee8bdc8c46e43fade85b5a2 with 1bcc7.. x0 x2 x4 x6, x1, x3, x5, x7.

The subproof is completed by applying H0.

Claim L2: x0 = x1

Apply L1 with λ x8 x9 . x0 = x9.

The subproof is completed by applying unknownprop_0cc92ec4e0e6dab19c106c8af8774f1cda73073bc0b40c4ccb1840d7af9b0b0d with x0, x2, x4, x6.

Apply and4I with x0 = x1, ∀ x8 . prim1 x8 x0 ⟶ x2 x8 = x3 x8, ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9, x6 = x7 leaving 4 subgoals.

The subproof is completed by applying L2.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Apply unknownprop_0fd914315b038e6cc084f062f01503a01c3efa5b54d8e88f9d1216b20fcbcdc3 with x0, x2, x4, x6, x8, λ x9 x10 . x10 = x3 x8 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x8 x1

Apply L2 with λ x9 x10 . prim1 x8 x9.

The subproof is completed by applying H3.

Apply H0 with λ x9 x10 . f482f.. (f482f.. x10 (4ae4a.. 4a7ef..)) x8 = x3 x8.

Let x9 of type ι → ι → ο be given.

Apply unknownprop_0fd914315b038e6cc084f062f01503a01c3efa5b54d8e88f9d1216b20fcbcdc3 with x1, x3, x5, x7, x8, λ x10 x11 . x9 x11 x10.

The subproof is completed by applying L4.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Let x9 of type ι be given.

Assume H4: prim1 x9 x0.

Apply unknownprop_022211655377b981fe7e19d29dd3c365db553d8685042e45a1cf4cc105d846fc with x0, x2, x4, x6, x8, x9, λ x10 x11 : ο . x11 = x5 x8 x9 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x8 x1

Apply L2 with λ x10 x11 . prim1 x8 x10.

The subproof is completed by applying H3.

Claim L6: prim1 x9 x1

Apply L2 with λ x10 x11 . prim1 x9 x10.

The subproof is completed by applying H4.

Apply H0 with λ x10 x11 . 2b2e3.. (f482f.. x11 (4ae4a.. (4ae4a.. 4a7ef..))) x8 x9 = x5 x8 x9.

Let x10 of type ο → ο → ο be given.

Apply unknownprop_022211655377b981fe7e19d29dd3c365db553d8685042e45a1cf4cc105d846fc with x1, x3, x5, x7, x8, x9, λ x11 x12 : ο . x10 x12 x11 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Apply unknownprop_e736d87fb1c82f1653455bc78e80c3d1deb331bc1d61d105d9782506c903a8fd with x0, x2, x4, x6, λ x8 x9 . x9 = x7.

Apply H0 with λ x8 x9 . f482f.. x9 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) = x7.

Let x8 of type ι → ι → ο be given.

The subproof is completed by applying unknownprop_e736d87fb1c82f1653455bc78e80c3d1deb331bc1d61d105d9782506c903a8fd with x1, x3, x5, x7, λ x9 x10 . x8 x10 x9.

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Proofgold Proof