Let x0 of type ι be given.
Let x1 of type (ι → ο) → ο be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ο be given.
Apply H0 with
λ x4 . x4 = c7ccc.. x0 x1 x2 x3 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 leaving 2 subgoals.
Let x4 of type ι be given.
Let x5 of type (ι → ο) → ο be given.
Let x6 of type ι → ι be given.
Let x7 of type ι → ο be given.
Apply unknownprop_e779883ca65505bf66d866e2b056994b7139927446d299ab2b394cb86a3341d1 with
x4,
x0,
x5,
x1,
x6,
x2,
x7,
x3,
∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3:
and (and (x4 = x0) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x4) ⟶ x5 x8 = x1 x8)) (∀ x8 . prim1 x8 x4 ⟶ x6 x8 = x2 x8).
Apply H3 with
(∀ x8 . prim1 x8 x4 ⟶ x7 x8 = x3 x8) ⟶ ∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0.
Assume H4:
and (x4 = x0) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x4) ⟶ x5 x8 = x1 x8).
Apply H4 with
(∀ x8 . prim1 x8 x4 ⟶ x6 x8 = x2 x8) ⟶ (∀ x8 . prim1 x8 x4 ⟶ x7 x8 = x3 x8) ⟶ ∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0.
Assume H5: x4 = x0.
Assume H6:
∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x4) ⟶ x5 x8 = x1 x8.
Assume H7:
∀ x8 . prim1 x8 x4 ⟶ x6 x8 = x2 x8.
Assume H8:
∀ x8 . prim1 x8 x4 ⟶ x7 x8 = x3 x8.
Apply H5 with
λ x8 x9 . ∀ x10 . prim1 x10 x8 ⟶ prim1 (x2 x10) x8.
Let x8 of type ι be given.
Apply H7 with
x8,
λ x9 x10 . prim1 x9 x4 leaving 2 subgoals.
The subproof is completed by applying H9.
Apply H1 with
x8.
The subproof is completed by applying H9.
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H1.