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 unknownprop_e032662aa31d0d7c6cd21c7ec8f978be1dee32c5d8585622c751063506e260f9 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
λ x10 x11 . x11 = x1 x9 (x1 x3 (x1 x7 (x1 x5 (x1 x4 (x1 x8 (x1 x6 x2)))))) leaving 11 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
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.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply H2 with
x2,
x9,
λ x10 x11 . x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x11))))) = x1 x9 (x1 x3 (x1 x7 (x1 x5 (x1 x4 (x1 x8 (x1 x6 x2)))))) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H10.
Apply unknownprop_dab1e802c87d7a5c9738afcfe324d44669b26e07d0d21638be3032cdf1ddc5d3 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x2 leaving 10 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H3.