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.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Assume H8: x0 x8.
Apply unknownprop_f8cd2cc9fa3527ece23bf769ef6e4562a4db8193110bd03a6fde0a07ab4c38b8 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x1 x7 x8,
λ x9 x10 . x10 = x1 (x1 (x2 x3 x7) (x2 x3 x8)) (x1 (x1 (x2 x4 x7) (x2 x4 x8)) (x1 (x1 (x2 x5 x7) (x2 x5 x8)) (x1 (x2 x6 x7) (x2 x6 x8)))) leaving 8 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.
Apply H0 with
x7,
x8 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H1 with
x3,
x7,
x8,
λ x9 x10 . x1 x10 (x1 (x2 x4 (x1 x7 x8)) (x1 (x2 x5 (x1 x7 x8)) (x2 x6 (x1 x7 x8)))) = x1 (x1 (x2 x3 x7) (x2 x3 x8)) (x1 (x1 (x2 x4 x7) (x2 x4 x8)) (x1 (x1 (x2 x5 x7) (x2 x5 x8)) (x1 (x2 x6 x7) (x2 x6 x8)))) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H1 with
x4,
x7,
x8,
λ x9 x10 . x1 (x1 (x2 x3 x7) (x2 x3 x8)) (x1 x10 (x1 (x2 x5 (x1 x7 x8)) (x2 x6 (x1 x7 x8)))) = x1 (x1 (x2 x3 x7) (x2 x3 x8)) (x1 (x1 (x2 x4 x7) (x2 x4 x8)) (x1 (x1 (x2 x5 x7) (x2 x5 x8)) (x1 (x2 x6 x7) (x2 x6 x8)))) leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H1 with
x5,
x7,
x8,
λ x9 x10 . x1 (x1 (x2 x3 x7) (x2 x3 x8)) (x1 (x1 (x2 x4 x7) (x2 x4 x8)) (x1 x10 (x2 x6 (x1 x7 ...)))) = ... leaving 4 subgoals.