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_2ce9a82c8ef9efc0240c60d5f07d019e2f7a44da8d6114bc529d6fb2d8f3a783 with
x0,
x1,
x5,
x6,
x7,
x8,
x9,
λ x10 x11 . x1 x2 (x1 x3 (x1 x4 x11)) = x1 x7 (x1 x4 (x1 x3 (x1 x8 (x1 x2 (x1 x9 (x1 x6 x5)))))) leaving 8 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.
The subproof is completed by applying H10.
Apply H2 with
x5,
x9,
λ x10 x11 . x1 x2 (x1 x3 (x1 x4 (x1 x6 (x1 x7 (x1 x8 x11))))) = x1 x7 (x1 x4 (x1 x3 (x1 x8 (x1 x2 (x1 x9 (x1 x6 x5)))))) leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H10.
Apply unknownprop_52767ea66f3bc01eeb7c04bf51f6f3263e8ab508685dfc5dc2aff77f9716e214 with
x0,
x1,
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x5 leaving 10 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 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 H6.