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 x2 (x1 x3 x4) = x1 x3 (x1 x2 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.
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 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.
Assume H8: x0 x7.
Assume H9: x0 x8.
Assume H10: x0 x9.
Apply H2 with
x8,
x9,
λ x10 x11 . x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x11))))) = x1 x6 (x1 x3 (x1 x4 (x1 x7 (x1 x2 (x1 x5 (x1 x9 x8)))))) leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Let x10 of type ι → ι → ο be given.
Apply unknownprop_9292b38a6c5629f8c27998c8d1db1edabaa4c59b037764df82cccb3d1506ee2e with
x0,
x1,
x6,
x3,
x4,
x7,
x2,
x5,
x1 x9 x8,
λ x11 x12 . x10 x12 x11 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
Apply H0 with
x9,
x8 leaving 2 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H9.