Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: SNo x0.

Assume H1: SNo x1.

Apply mul_SNo_prop_1 with x0, x1, SNo (mul_SNo x0 x1) leaving 3 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Assume H2: SNo (mul_SNo x0 x1).

Assume H3: ∀ x2 . x2 ∈ SNoL x0 ⟶ ∀ x3 . x3 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)).

Assume H4: ∀ x2 . x2 ∈ SNoR x0 ⟶ ∀ x3 . x3 ∈ SNoR x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)).

Assume H5: ∀ x2 . x2 ∈ SNoL x0 ⟶ ∀ x3 . x3 ∈ SNoR x1 ⟶ SNoLt (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)) (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)).

Assume H6: ∀ x2 . x2 ∈ SNoR x0 ⟶ ∀ x3 . x3 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)) (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)).

The subproof is completed by applying H2.

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Proofgold Proof