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 → ο be given.

Let x5 of type ι → ι → ο be given.

Assume H0: b5cc3.. x0 x2 x4 = b5cc3.. x1 x3 x5.

Claim L1: x1 = f482f.. (b5cc3.. x0 x2 x4) 4a7ef..

Apply unknownprop_f31eb2228dfee2fe4dd27f9052dc9bb4b4e8eea40ff1d87c4fda262a65a942a4 with b5cc3.. x0 x2 x4, x1, x3, x5.

The subproof is completed by applying H0.

Claim L2: x0 = x1

Apply L1 with λ x6 x7 . x0 = x7.

The subproof is completed by applying unknownprop_2e4ad16a724aa9ecb1f2d0714afea104f57cf750321b796a86385476dc14b16d with x0, x2, x4.

Apply and3I with x0 = x1, ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7, ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x6 x7 = x5 x6 x7 leaving 3 subgoals.

The subproof is completed by applying L2.

Let x6 of type ι be given.

Assume H3: prim1 x6 x0.

Let x7 of type ι be given.

Assume H4: prim1 x7 x0.

Apply unknownprop_15cd99d9ca461ad2237964118d360f2606f8cb1d2c65d040d581982ac836b5ae with x0, x2, x4, x6, x7, λ x8 x9 . x9 = x3 x6 x7 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x6 x1

Apply L2 with λ x8 x9 . prim1 x6 x8.

The subproof is completed by applying H3.

Claim L6: prim1 x7 x1

Apply L2 with λ x8 x9 . prim1 x7 x8.

The subproof is completed by applying H4.

Apply H0 with λ x8 x9 . e3162.. (f482f.. x9 (4ae4a.. 4a7ef..)) x6 x7 = x3 x6 x7.

Let x8 of type ι → ι → ο be given.

Apply unknownprop_15cd99d9ca461ad2237964118d360f2606f8cb1d2c65d040d581982ac836b5ae with x1, x3, x5, x6, x7, λ x9 x10 . x8 x10 x9 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x6 of type ι be given.

Assume H3: prim1 x6 x0.

Let x7 of type ι be given.

Assume H4: prim1 x7 x0.

Apply unknownprop_1b1870aad03d524430ecdbf45bff93c05749be68abe1a80eafa30ff7d1c8a1e5 with x0, x2, x4, x6, x7, λ x8 x9 : ο . x9 = x5 x6 x7 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x6 x1

Apply L2 with λ x8 x9 . prim1 x6 x8.

The subproof is completed by applying H3.

Claim L6: prim1 x7 x1

Apply L2 with λ x8 x9 . prim1 x7 x8.

The subproof is completed by applying H4.

Apply H0 with λ x8 x9 . 2b2e3.. (f482f.. x9 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 = x5 x6 x7.

Let x8 of type ο → ο → ο be given.

Apply unknownprop_1b1870aad03d524430ecdbf45bff93c05749be68abe1a80eafa30ff7d1c8a1e5 with x1, x3, x5, x6, x7, λ x9 x10 : ο . x8 x10 x9 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

■

Proofgold Proof