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 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H2: ∀ 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.
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.
Apply unknownprop_bfc1cea51a42a4a32469869e8448699c1d59a9077d0b467664f278e66191db35 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x1 x9 x10,
λ x11 x12 . x12 = x1 (x1 (x2 x3 x9) (x2 x3 x10)) (x1 (x1 (x2 x4 x9) (x2 x4 x10)) (x1 (x1 (x2 x5 x9) (x2 x5 x10)) (x1 (x1 (x2 x6 x9) (x2 x6 x10)) (x1 (x1 (x2 x7 x9) (x2 x7 x10)) (x1 (x2 x8 x9) (x2 x8 x10)))))) leaving 10 subgoals.
The subproof is completed by applying H0.
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.
Apply H0 with
x9,
x10 leaving 2 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply H1 with
x3,
x9,
x10,
λ x11 x12 . x1 x12 (x1 (x2 x4 (x1 x9 x10)) (x1 (x2 x5 (x1 x9 x10)) (x1 (x2 x6 (x1 x9 x10)) (x1 (x2 x7 (x1 x9 x10)) (x2 x8 (x1 x9 x10)))))) = x1 (x1 (x2 x3 x9) (x2 x3 x10)) (x1 (x1 (x2 x4 x9) (x2 x4 x10)) (x1 (x1 (x2 x5 x9) (x2 x5 x10)) (x1 (x1 (x2 x6 x9) (x2 x6 x10)) (x1 (x1 (x2 x7 x9) (x2 x7 x10)) (x1 (x2 x8 x9) (x2 x8 x10)))))) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply H1 with
x4,
x9,
x10,
λ x11 x12 . x1 (x1 (x2 x3 x9) (x2 x3 x10)) (x1 x12 (x1 (x2 x5 (x1 x9 x10)) (x1 (x2 x6 (x1 x9 x10)) (x1 (x2 x7 (x1 x9 ...)) ...)))) = ... leaving 4 subgoals.