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_2ce9a82c8ef9efc0240c60d5f07d019e2f7a44da8d6114bc529d6fb2d8f3a783 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
λ x9 x10 . x1 x2 (x1 x3 x10) = x1 x8 (x1 x6 (x1 x5 (x1 x7 (x1 x2 (x1 x3 x4))))) leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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.
Apply H2 with
x4,
x8,
λ x9 x10 . x1 x2 (x1 x3 (x1 x5 (x1 x6 (x1 x7 x10)))) = x1 x8 (x1 x6 (x1 x5 (x1 x7 (x1 x2 (x1 x3 x4))))) leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H9.
Apply unknownprop_469618a9094d7b9e8d947e0521ab963bcbed5a905b148b2f3738d3cefb906630 with
x0,
x1,
x2,
x3,
x5,
x6,
x7,
x8,
x4 leaving 9 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 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 H5.