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.
Let x9 of type ι be given.
Let x10 of type ι be given.
Let x11 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.
Assume H9: x0 x9.
Assume H10: x0 x10.
Assume H11: x0 x11.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11)))))),
λ x12 x13 . x13 = x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x2 x11)))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11))))) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11)))) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x7 (x1 x8 (x1 x9 (x1 x10 x11))) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x8 (x1 x9 (x1 x10 x11)) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x8,
x1 x9 (x1 x10 x11) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with
x9,
x1 x10 x11 leaving 2 subgoals.
The subproof is completed by applying H9.
Apply H0 with
x10,
x11 leaving 2 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
set y12 to be x1 x3 (x1 x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 ... ...)))))))
set y13 to be x2 x4 (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x9 (x2 x10 (x2 x11 (x2 x3 y12))))))))
Claim L12: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H12: x14 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 y12 (x3 x4 y13))))))))).
set y15 to be λ x15 . x14
Apply unknownprop_f348ca4750d331c757263d6f373e891336185bf4dbd51b158fe78a2ebc2cf398 with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
y12,
y13,
λ x16 x17 . y15 (x3 x5 x16) (x3 x5 x17) leaving 12 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
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.
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 H11.
The subproof is completed by applying H12.
Let x14 of type ι → ι → ο be given.
Apply L12 with
λ x15 . x14 x15 y13 ⟶ x14 y13 x15.
Assume H13: x14 y13 y13.
The subproof is completed by applying H13.