Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: SNo x0.

Assume H1: SNo x1.

set y2 to be mul_SNo x0 (minus_SNo x1)

set y3 to be minus_SNo (mul_SNo x1 y2)

Claim L2: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2

Let x4 of type ι → ο be given.

Assume H2: x4 (minus_SNo (mul_SNo y2 y3)).

Apply mul_SNo_com with y2, minus_SNo y3, λ x5 . x4 leaving 3 subgoals.

The subproof is completed by applying H0.

Apply SNo_minus_SNo with y3.

The subproof is completed by applying H1.

Apply mul_SNo_minus_distrL with y3, y2, λ x5 . x4 leaving 3 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H0.

set y5 to be minus_SNo (mul_SNo y3 y2)

set y6 to be minus_SNo (mul_SNo y3 x4)

Claim L3: ∀ x7 : ι → ο . x7 y6 ⟶ x7 y5

Let x7 of type ι → ο be given.

Assume H3: x7 (minus_SNo (mul_SNo x4 y5)).

set y8 to be λ x8 . x7

Apply mul_SNo_com with y5, x4, λ x9 x10 . y8 (minus_SNo x9) (minus_SNo x10) leaving 3 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H0.

The subproof is completed by applying H3.

set y7 to be λ x7 . y6

Apply L3 with λ x8 . y7 x8 y6 ⟶ y7 y6 x8 leaving 2 subgoals.

Assume H4: y7 y6 y6.

The subproof is completed by applying H4.

The subproof is completed by applying L3.

Let x4 of type ι → ι → ο be given.

Apply L2 with λ x5 . x4 x5 y3 ⟶ x4 y3 x5.

Assume H3: x4 y3 y3.

The subproof is completed by applying H3.

■

Proofgold Proof