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.

Assume H0: 971d3.. x0 x2 x4 = 971d3.. x1 x3 x5.

Claim L1: x1 = f482f.. (971d3.. x0 x2 x4) 4a7ef..

Apply unknownprop_94967784b176f60b5bf438df3a2971cde72407446df861d384c2aae63aae54a8 with 971d3.. 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_0de249c9a6cc0d3bcd5cc89147ea799acb4f233823c55edb760ffe9d4ffc7a91 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 ⟶ x4 x6 = x5 x6 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_6a5674efb6b0d571fb24cd5f7a30b471eec1bdac0d350bb303a07726f30cd6bf 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_6a5674efb6b0d571fb24cd5f7a30b471eec1bdac0d350bb303a07726f30cd6bf 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.

Apply unknownprop_8107478b497c3b04aa44623f19e37dac344de98a93384327865b5df8b8117888 with x0, x2, x4, x6, λ x7 x8 . x8 = x5 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.. (4ae4a.. 4a7ef..))) x6 = x5 x6.

Let x7 of type ι → ι → ο be given.

Apply unknownprop_8107478b497c3b04aa44623f19e37dac344de98a93384327865b5df8b8117888 with x1, x3, x5, x6, λ x8 x9 . x7 x9 x8.

The subproof is completed by applying L4.

■

Proofgold Proof