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.
Let x8 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.
Assume H8: x0 x8.
Apply unknownprop_f0b76402e77112232d36cbd146a3e3efd40fdf823d5b935fe896a6fb8918a817 with
x0,
x1,
x6,
x7,
x3,
x4,
x5,
x8,
λ x9 x10 . x1 x2 x9 = x1 x6 (x1 x4 (x1 x5 (x1 x7 (x1 x2 (x1 x3 x8))))) leaving 9 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 H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
Apply H1 with
x2,
x6,
x1 x7 (x1 x3 (x1 x4 (x1 x5 x8))),
λ x9 x10 . x10 = x1 x6 (x1 x4 (x1 x5 (x1 x7 (x1 x2 (x1 x3 x8))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x3 (x1 x4 (x1 x5 x8)) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x3,
x1 x4 (x1 x5 x8) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 x8 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x8 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
set y9 to be x1 x6 (x1 x2 (x1 x7 (x1 x3 (x1 x4 (x1 x5 x8)))))
set y10 to be x2 x7 (x2 x5 (x2 x6 (x2 x8 (x2 x3 (x2 x4 y9)))))
Claim L9: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H9: x11 (x3 x8 (x3 x6 (x3 x7 (x3 y9 (x3 x4 (x3 x5 y10)))))).
set y12 to be λ x12 . x11
set y13 to be λ x13 x14 . y12 (x3 x8 x13) (x3 x8 x14)
Apply unknownprop_511c4d305aaefbf21185ab89172f8ae3a48452ea9c12d4f653677a8db75f4834 with
x2,
x3,
x6,
x7,
y9,
x4,
x5,
y10,
λ x14 x15 . y13 x15 x14 leaving 9 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 H5.
The subproof is completed by applying H7.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Let x11 of type ι → ι → ο be given.
Apply L9 with
λ x12 . x11 x12 y10 ⟶ x11 y10 x12.
Assume H10: x11 y10 y10.
The subproof is completed by applying H10.