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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.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNoLt 0 x1.
Assume H4: SNoLt (mul_SNo x2 x1) x0.
Apply mul_SNo_oneR with x2, λ x3 x4 . SNoLt x3 (div_SNo x0 x1) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply recip_SNo_invR with x1, λ x3 x4 . SNoLt (mul_SNo x2 x3) (div_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H5: x1 = 0.
Apply SNoLt_irref with x1.
Apply H5 with λ x3 x4 . SNoLt x4 x1.
The subproof is completed by applying H3.
Apply mul_SNo_assoc with x2, x1, recip_SNo x1, λ x3 x4 . SNoLt x4 (mul_SNo x0 (recip_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_recip_SNo with x1.
The subproof is completed by applying H1.
Apply pos_mul_SNo_Lt' with mul_SNo x2 x1, x0, recip_SNo x1 leaving 5 subgoals.
Apply SNo_mul_SNo with x2, x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply SNo_recip_SNo with x1.
The subproof is completed by applying H1.
Apply recip_SNo_of_pos_is_pos with x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.