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Proofgold Proof

pf
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 x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNo x3.
Apply mul_SNo_assoc with mul_SNo x0 x1, x2, x3, λ x4 x5 . x5 = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3) leaving 4 subgoals.
Apply SNo_mul_SNo with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply mul_SNo_com_3b_1_2 with x0, x1, x2, λ x4 x5 . mul_SNo x5 x3 = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ιιο be given.
Apply mul_SNo_assoc with mul_SNo x0 x2, x1, x3, λ x5 x6 . x4 x6 x5 leaving 3 subgoals.
Apply SNo_mul_SNo with x0, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H3.