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).
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.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Apply unknownprop_2ce9a82c8ef9efc0240c60d5f07d019e2f7a44da8d6114bc529d6fb2d8f3a783 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 . x1 x2 x9 = x1 x4 (x1 x5 (x1 x2 (x1 x6 (x1 x3 x7)))) 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 H1 with
x2,
x4,
x1 x5 (x1 x6 (x1 x3 x7)),
λ x8 x9 . x9 = x1 x4 (x1 x5 (x1 x2 (x1 x6 (x1 x3 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x3 x7) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x3 x7 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x3,
x7 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H7.
set y8 to be x1 x4 (x1 x2 (x1 x5 (x1 x6 (x1 x3 x7))))
set y9 to be x2 x5 (x2 x6 (x2 x3 (x2 x7 (x2 x4 y8))))
Claim L8: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H8: x10 (x3 x6 (x3 x7 (x3 x4 (x3 y8 (x3 x5 y9))))).
set y11 to be λ x11 . x10
Apply H1 with
x4,
x7,
x3 y8 (x3 x5 y9),
λ x12 x13 . y11 (x3 x6 x12) (x3 x6 x13) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply H0 with
y8,
x3 x5 y9 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x5,
y9 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Let x10 of type ι → ι → ο be given.
Apply L8 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H9: x10 y9 y9.
The subproof is completed by applying H9.