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.
Let x12 of type ι be given.
Let x13 of type ι be given.
Let x14 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.
Assume H12: x0 x12.
Assume H13: x0 x13.
Assume H14: x0 x14.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14))))))))),
λ x15 x16 . x16 = x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 (x1 x2 x14))))))))))) 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 (x1 x11 (x1 x12 (x1 x13 x14)))))))) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14))))))) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14)))))) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14))))) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x8,
x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14)))) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with
x9,
x1 x10 (x1 x11 (x1 x12 (x1 x13 x14))) leaving 2 subgoals.
The subproof is completed by applying H9.
Apply H0 with
x10,
x1 x11 (x1 x12 (x1 x13 x14)) leaving 2 subgoals.
The subproof is completed by applying H10.
Apply H0 with
x11,
x1 x12 (x1 x13 x14) leaving 2 subgoals.
The subproof is completed by applying H11.
Apply H0 with
x12,
x1 x13 x14 leaving 2 subgoals.
The subproof is completed by applying H12.
Apply H0 with
x13,
x14 leaving 2 subgoals.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
set y15 to be ...
set y16 to be ...
Claim L15: ∀ x17 : ι → ο . x17 y16 ⟶ x17 y15
Let x17 of type ι → ο be given.
Assume H15: x17 (x3 x5 ...).
Let x17 of type ι → ι → ο be given.
Apply L15 with
λ x18 . x17 x18 y16 ⟶ x17 y16 x18.
Assume H16: x17 y16 y16.
The subproof is completed by applying H16.