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 (x1 x2 x3) x4 = x1 x2 (x1 x3 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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Apply unknownprop_c1138762243f92f9929761f5486cb68feb113061102de6c98d30b7c79400b203 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 . x7 = x1 x3 (x1 x2 (x1 x4 (x1 x5 x6))) leaving 8 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.
Apply unknownprop_c1138762243f92f9929761f5486cb68feb113061102de6c98d30b7c79400b203 with
x0,
x1,
x3,
x2,
x4,
x5,
x6,
λ x7 x8 . x1 (x1 x2 x3) (x1 x4 (x1 x5 x6)) = x7 leaving 8 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 H3.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
set y7 to be x1 (x1 x2 x3) (x1 x4 (x1 x5 x6))
set y8 to be x2 (x2 x4 x3) (x2 x5 (x2 x6 y7))
Claim L8: ∀ x9 : ι → ο . x9 y8 ⟶ x9 y7
Let x9 of type ι → ο be given.
Assume H8: x9 (x3 (x3 x5 x4) (x3 x6 (x3 y7 y8))).
set y10 to be λ x10 . x9
Apply H2 with
x4,
x5,
λ x11 x12 . y10 (x3 x11 (x3 x6 (x3 y7 y8))) (x3 x12 (x3 x6 (x3 y7 y8))) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
Let x9 of type ι → ι → ο be given.
Apply L8 with
λ x10 . x9 x10 y8 ⟶ x9 y8 x10.
Assume H9: x9 y8 y8.
The subproof is completed by applying H9.