Let x0 of type ι be given.

Let x1 of type ι → ι → ι be given.

Assume H0: ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0.

Let x2 of type ι → ι → ι be given.

Assume H1: ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0.

Let x3 of type ι be given.

Assume H2: prim1 x3 x0.

Let x4 of type ι → ο be given.

Assume H3: ∀ x5 . ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x5 ⟶ ∀ x8 . prim1 x8 x5 ⟶ prim1 (x6 x7 x8) x5) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x5 ⟶ ∀ x9 . prim1 x9 x5 ⟶ prim1 (x7 x8 x9) x5) ⟶ ∀ x8 . prim1 x8 x5 ⟶ x4 (e707a.. x5 x6 x7 x8).

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

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

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Proofgold Proof