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.
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.
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 x16))))))))))),
x17,
λ x18 x19 . x19 = x1 (x2 x3 x17) (x1 (x2 x4 x17) (x1 (x2 x5 x17) (x1 (x2 x6 x17) (x1 (x2 x7 x17) (x1 (x2 x8 x17) (x1 (x2 x9 x17) (x1 (x2 x10 x17) (x1 (x2 x11 x17) (x1 (x2 x12 x17) (x1 (x2 x13 x17) (x1 (x2 x14 x17) (x1 (x2 x15 x17) (x2 x16 x17))))))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_afb4cfc9f10424b1b5cf811579ab414fa5effe9e0015718aad3c27c7fa587bda with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
x15,
x16 leaving 14 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.
set y18 to be ...
set y19 to be ...
Claim L17: ∀ x20 : ι → ο . x20 y19 ⟶ x20 y18
Let x20 of type ι → ο be given.
Assume H17: x20 (x3 (x4 x5 y19) (x3 (x4 x6 y19) (x3 (x4 x7 y19) (x3 (x4 x8 y19) (x3 (x4 x9 y19) (x3 (x4 x10 y19) (x3 (x4 x11 y19) (x3 (x4 x12 y19) (x3 (x4 x13 y19) (x3 (x4 x14 y19) (x3 (x4 x15 y19) (x3 (x4 x16 y19) (x3 (x4 x17 y19) (x4 y18 y19)))))))))))))).
Let x20 of type ι → ι → ο be given.
Apply L17 with
λ x21 . x20 x21 y19 ⟶ x20 y19 x21.
Assume H18: x20 y19 y19.
The subproof is completed by applying H18.