Let x0 of type ι be given.

Let x1 of type (ι → ο) → ο be given.

Let x2 of type ι → ι → ι be given.

Assume H0: ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0.

Let x3 of type ι → ι be given.

Assume H1: ∀ x4 . prim1 x4 x0 ⟶ prim1 (x3 x4) x0.

Let x4 of type ι be given.

Assume H2: prim1 x4 x0.

Let x5 of type ι → ο be given.

Assume H3: ∀ x6 . ∀ x7 : (ι → ο) → ο . ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x6 ⟶ ∀ x10 . prim1 x10 x6 ⟶ prim1 (x8 x9 x10) x6) ⟶ ∀ x9 : ι → ι . (∀ x10 . prim1 x10 x6 ⟶ prim1 (x9 x10) x6) ⟶ ∀ x10 . prim1 x10 x6 ⟶ x5 (a0d6b.. x6 x7 x8 x9 x10).

Apply H3 with x0, x1, x2, x3, x4 leaving 3 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

■

Proofgold Proof