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 (x1 x2 x3) x4 = x1 x2 (x1 x3 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.
Let x15 of type ι be given.
Let x16 of type ι be given.
Let x17 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.
Assume H15: x0 x15.
Assume H16: x0 x16.
Assume H17: x0 x17.
Apply unknownprop_675d728464e1b9c568c7db48bf4ccd2afa35b2188ec42cca4d748833029fdcb3 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x1 x14 x15,
x16,
x17,
λ x18 x19 . x19 = x1 x2 (x1 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)))))))))))))) leaving 18 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 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.
Apply H0 with
x14,
x15 leaving 2 subgoals.
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 y18 to be ...
set y19 to be ...
Claim L18: ∀ x20 : ι → ο . x20 y19 ⟶ x20 y18
Let x20 of type ι → ο be given.
Assume H18: x20 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 x13 (x3 x14 (x3 x15 (x3 x16 (x3 x17 (x3 y18 y19))))))))))))))).
set y21 to be ...
set y22 to be ...
set y23 to be ...
Claim L19: ∀ x24 : ι → ο . x24 y23 ⟶ x24 y22
Let x24 of type ι → ο be given.
Assume H19: x24 (x5 x7 (x5 x8 (x5 x9 (x5 x10 (x5 x11 (x5 x12 (x5 x13 (x5 x14 (x5 x15 (x5 x16 (x5 x17 (x5 y18 (x5 y19 (x5 x20 y21)))))))))))))).
set y25 to be ...
set y26 to be ...
set y27 to be ...
set y28 to be ...
Apply L20 with
λ x29 . y28 x29 y27 ⟶ y28 y27 x29 leaving 2 subgoals.
Assume H21: y28 y27 y27.
The subproof is completed by applying H21.
set y24 to be λ x24 x25 . y23 (x5 x6 x24) (x5 x6 x25)
Apply L19 with
λ x25 . y24 x25 y23 ⟶ y24 y23 x25 leaving 2 subgoals.
Assume H20: y24 y23 y23.
The subproof is completed by applying H20.
The subproof is completed by applying L19.
Let x20 of type ι → ι → ο be given.
Apply L18 with
λ x21 . x20 x21 y19 ⟶ x20 y19 x21.
Assume H19: x20 y19 y19.
The subproof is completed by applying H19.