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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNoLt 0 x0.
Assume H3: SNoLt 0 x1.
Assume H4: mul_SNo x0 x0 = mul_SNo x1 x1.
Apply SNoLt_trichotomy_or_impred with x0, x1, x0 = x1 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H5: SNoLt x0 x1.
Apply FalseE with x0 = x1.
Apply SNoLt_irref with mul_SNo x0 x0.
Apply H4 with λ x2 x3 . SNoLt (mul_SNo x0 x0) x3.
Apply pos_mul_SNo_Lt2 with x0, x0, x1, x1 leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
The subproof is completed by applying H5.
Assume H5: x0 = x1.
The subproof is completed by applying H5.
Assume H5: SNoLt x1 x0.
Apply FalseE with x0 = x1.
Apply SNoLt_irref with mul_SNo x0 x0.
Apply H4 with λ x2 x3 . SNoLt x3 (mul_SNo x0 x0).
Apply pos_mul_SNo_Lt2 with x1, x1, x0, x0 leaving 8 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H5.