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.

Assume H0: 73737.. x0 x2 x4 = 73737.. x1 x3 x5.

Claim L1: x1 = f482f.. (73737.. x0 x2 x4) 4a7ef..

Apply unknownprop_ff14fe4b2b5ebb9a364ac3d2b72652c556f881650361e9e04e15eb4f78727558 with 73737.. 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_56fff63ae4aaffa24f98b7fb5e034afaf3bd48ea6cf337512ab94a8fb832e90d with x0, x2, x4.

Apply and3I with x0 = x1, ∀ x6 . prim1 x6 x0 ⟶ x2 x6 = x3 x6, ∀ 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.

Apply unknownprop_0eba4a1e8a6eb85bd93458d64288a202dddf6558775b796d073829b1ed4aed10 with x0, x2, x4, x6, λ x7 x8 . x8 = x3 x6 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x6 x1

Apply L2 with λ x7 x8 . prim1 x6 x7.

The subproof is completed by applying H3.

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

Let x7 of type ι → ι → ο be given.

Apply unknownprop_0eba4a1e8a6eb85bd93458d64288a202dddf6558775b796d073829b1ed4aed10 with x1, x3, x5, x6, λ x8 x9 . x7 x9 x8.

The subproof is completed by applying L4.

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_d3745c59fd830feec1cf172211f6f2a659a425d7b1b42063659cffa5062c6422 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_d3745c59fd830feec1cf172211f6f2a659a425d7b1b42063659cffa5062c6422 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