Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: SNo x0.

Assume H1: SNo x1.

Assume H2: SNo x2.

set y3 to be div_SNo (mul_SNo x2 x0) x1

Claim L3: ∀ x4 : ι → ο . x4 y3 ⟶ x4 (mul_SNo x2 (div_SNo x0 x1))

Let x4 of type ι → ο be given.

Assume H3: x4 (div_SNo (mul_SNo y3 x1) x2).

Apply mul_SNo_com with y3, div_SNo x1 x2, λ x5 . x4 leaving 3 subgoals.

The subproof is completed by applying H2.

Apply SNo_div_SNo with x1, x2 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Apply mul_div_SNo_R with x1, x2, y3, λ x5 . x4 leaving 4 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

set y5 to be div_SNo (mul_SNo y3 x1) x2

Claim L4: ∀ x6 : ι → ο . x6 y5 ⟶ x6 (div_SNo (mul_SNo x1 y3) x2)

Let x6 of type ι → ο be given.

Apply mul_SNo_com with x2, x4, λ x7 x8 . (λ x9 . x6) (div_SNo x7 y3) (div_SNo x8 y3) leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H2.

set y6 to be λ x6 . y5

Apply L4 with λ x7 . y6 x7 y5 ⟶ y6 y5 x7 leaving 2 subgoals.

Assume H5: y6 y5 y5.

The subproof is completed by applying H5.

The subproof is completed by applying L4.

Let x4 of type ι → ι → ο be given.

Apply L3 with λ x5 . x4 x5 y3 ⟶ x4 y3 x5.

Assume H4: x4 y3 y3.

The subproof is completed by applying H4.

■

Proofgold Proof