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.
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.
Apply unknownprop_6df806693864a23a378ddbca02cda4bb4bc233ff1daa8914d51c06eb72ff2550 with
x0,
x1,
x5,
x6,
x7,
x8,
λ x9 x10 . x1 x2 (x1 x3 (x1 x4 x10)) = x1 x8 (x1 x3 (x1 x4 (x1 x7 (x1 x6 (x1 x2 x5))))) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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.
Apply H2 with
x5,
x8,
λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x6 (x1 x7 x10)))) = x1 x8 (x1 x3 (x1 x4 (x1 x7 (x1 x6 (x1 x2 x5))))) leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H9.
Let x9 of type ι → ι → ο be given.
Apply unknownprop_131535c0a52fa29508e383421dcfa3816fae038e4c6b01758d899ceeeb8f06a4 with
x0,
x1,
x8,
x3,
x4,
x7,
x6,
x2,
x5,
λ x10 x11 . x9 x11 x10 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H9.
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 H7.
The subproof is completed by applying H3.
The subproof is completed by applying H6.