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 → ο be given.

Assume H2: ∀ x5 . ∀ x6 : (ι → ο) → ο . ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x5 ⟶ ∀ x9 . prim1 x9 x5 ⟶ prim1 (x7 x8 x9) x5) ⟶ ∀ x8 : ι → ι . (∀ x9 . prim1 x9 x5 ⟶ prim1 (x8 x9) x5) ⟶ x4 (21805.. x5 x6 x7 x8).

Apply H2 with x0, x1, x2, x3 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

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Proofgold Proof