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.
Let x19 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.
Assume H18: x0 x19.
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 (x1 x17 x18))))))))))))),
x19,
λ x20 x21 . x21 = x1 (x2 x3 x19) (x1 (x2 x4 x19) (x1 (x2 x5 x19) (x1 (x2 x6 x19) (x1 (x2 x7 x19) (x1 (x2 x8 x19) (x1 (x2 x9 x19) (x1 (x2 x10 x19) (x1 (x2 x11 x19) (x1 (x2 x12 x19) (x1 (x2 x13 x19) (x1 (x2 x14 x19) (x1 (x2 x15 x19) (x1 (x2 x16 x19) (x1 (x2 x17 x19) (x2 x18 x19))))))))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_ef97c49aecb41782ce54e00ef73e10bfe2798e4a7d1710ef62b52cff2f671e25 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
x15,
x16,
x17,
x18 leaving 16 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.
The subproof is completed by applying H18.
set y20 to be ...
set y21 to be ...
Claim L19: ∀ x22 : ι → ο . x22 y21 ⟶ x22 y20
Let x22 of type ι → ο be given.
Assume H19: x22 (x3 (x4 x5 y21) (x3 (x4 x6 y21) (x3 (x4 x7 y21) (x3 (x4 x8 y21) (x3 (x4 x9 y21) (x3 (x4 x10 y21) (x3 (x4 x11 y21) (x3 (x4 x12 y21) (x3 (x4 x13 ...) ...))))))))).
Let x22 of type ι → ι → ο be given.
Apply L19 with
λ x23 . x22 x23 y21 ⟶ x22 y21 x23.
Assume H20: x22 y21 y21.
The subproof is completed by applying H20.