Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ο be given.
Apply H0 with
λ x3 . x3 = b5cc3.. x0 x1 x2 ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 leaving 2 subgoals.
Let x3 of type ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ι → ο be given.
Apply unknownprop_81f50ce581f2df7efe78405976e8abc06e39d88757c8777dde0290302ef69e6c with
x3,
x0,
x4,
x1,
x5,
x2,
∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ prim1 (x1 x6 x7) x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3:
and (x3 = x0) (∀ x6 . prim1 x6 x3 ⟶ ∀ x7 . prim1 x7 x3 ⟶ x4 x6 x7 = x1 x6 x7).
Apply H3 with
(∀ x6 . prim1 x6 x3 ⟶ ∀ x7 . prim1 x7 x3 ⟶ x5 x6 x7 = x2 x6 x7) ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ prim1 (x1 x6 x7) x0.
Assume H4: x3 = x0.
Assume H5:
∀ x6 . prim1 x6 x3 ⟶ ∀ x7 . prim1 x7 x3 ⟶ x4 x6 x7 = x1 x6 x7.
Assume H6:
∀ x6 . prim1 x6 x3 ⟶ ∀ x7 . prim1 x7 x3 ⟶ x5 x6 x7 = x2 x6 x7.
Apply H4 with
λ x6 x7 . ∀ x8 . prim1 x8 x6 ⟶ ∀ x9 . prim1 x9 x6 ⟶ prim1 (x1 x8 x9) x6.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply H5 with
x6,
x7,
λ x8 x9 . prim1 x8 x3 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H1 with
x6,
x7 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Let x3 of type ι → ι → ο be given.
The subproof is completed by applying H1.