Let x0 of type ι → ο be given.

Let x1 of type ι → ι → ι be given.

Assume H0: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3).

Assume H1: ∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 (x1 x2 x3) x4 = x1 x2 (x1 x3 x4).

Assume H2: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2.

Let x2 of type ι be given.

Let x3 of type ι be given.

Let x4 of type ι be given.

Let x5 of type ι be given.

Let x6 of type ι be given.

Assume H3: x0 x2.

Assume H4: x0 x3.

Assume H5: x0 x4.

Assume H6: x0 x5.

Assume H7: x0 x6.

Apply H2 with x2, x1 x3 (x1 x4 (x1 x5 x6)), λ x7 x8 . x8 = x1 x3 (x1 x4 (x1 x5 (x1 x6 x2))) leaving 3 subgoals.

The subproof is completed by applying H3.

Apply unknownprop_b48d4480a5526e51a91293fec1b0b9440be4280265441ce358bda14cced12479 with x0, x1, x3, x4, x5, x6 leaving 5 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H4.

The subproof is completed by applying H5.

The subproof is completed by applying H6.

The subproof is completed by applying H7.

Apply unknownprop_c3af028adb093906f05778689ae9fc77f45e6f5023196e33f45d38dc06071320 with x0, x1, x3, x4, x5, x6, x2 leaving 7 subgoals.