Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Assume H0: SNo x1.

Assume H1: SNo x2.

Assume H2: SNo x3.

set y4 to be mul_SNo x0 (mul_SNo x2 (mul_SNo x1 x3))

Claim L3: ∀ x5 : ι → ο . x5 y4 ⟶ x5 (mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3)))

Let x5 of type ι → ο be given.

Apply mul_SNo_com_3_0_1 with x2, x3, y4, λ x6 x7 . (λ x8 . x5) (mul_SNo x1 x6) (mul_SNo x1 x7) leaving 3 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

Let x5 of type ι → ι → ο be given.

Apply L3 with λ x6 . x5 x6 y4 ⟶ x5 y4 x6.

Assume H4: x5 y4 y4.

The subproof is completed by applying H4.

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Proofgold Proof