Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
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.
Let x15 of type ι be given.
Let x16 of type ι be given.
Let x17 of type ι be given.
Let x18 of type ι be given.
Assume H2: x0 x3.
Assume H3: x0 x4.
Assume H4: x0 x5.
Assume H5: x0 x6.
Assume H6: x0 x7.
Assume H7: x0 x8.
Assume H8: x0 x9.
Assume H9: x0 x10.
Assume H10: x0 x11.
Assume H11: x0 x12.
Assume H12: x0 x13.
Assume H13: x0 x14.
Assume H14: x0 x15.
Assume H15: x0 x16.
Assume H16: x0 x17.
Assume H17: x0 x18.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 (x1 x14 (x1 x15 (x1 x16 x17)))))))))))),
x18,
λ x19 x20 . x20 = x1 (x2 x3 x18) (x1 (x2 x4 x18) (x1 (x2 x5 x18) (x1 (x2 x6 x18) (x1 (x2 x7 x18) (x1 (x2 x8 x18) (x1 (x2 x9 x18) (x1 (x2 x10 x18) (x1 (x2 x11 x18) (x1 (x2 x12 x18) (x1 (x2 x13 x18) (x1 (x2 x14 x18) (x1 (x2 x15 x18) (x1 (x2 x16 x18) (x2 x17 x18)))))))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_409e76161cbc9a3f73980a052c8613def869b897b75a997814dec31668923f11 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
x15,
x16,
x17 leaving 15 subgoals.
The subproof is completed by applying H0.
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.
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.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
The subproof is completed by applying H17.
set y19 to be ...
set y20 to be ...
Claim L18: ∀ x21 : ι → ο . x21 y20 ⟶ x21 y19
Let x21 of type ι → ο be given.
Assume H18: x21 (x3 (x4 x5 y20) (x3 (x4 x6 y20) (x3 (x4 x7 y20) (x3 (x4 x8 y20) (x3 (x4 x9 y20) (x3 (x4 x10 y20) (x3 (x4 x11 y20) (x3 (x4 x12 y20) (x3 (x4 x13 y20) (x3 (x4 x14 y20) (x3 (x4 x15 y20) (x3 ... ...)))))))))))).
Let x21 of type ι → ι → ο be given.
Apply L18 with
λ x22 . x21 x22 y20 ⟶ x21 y20 x22.
Assume H19: x21 y20 y20.
The subproof is completed by applying H19.