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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: x0 = 0 ⟶ ∀ x3 : ο . x3.
Assume H2: SNo x1.
Assume H3: SNo x2.
Assume H4: mul_SNo x0 x1 = mul_SNo x0 x2.
Apply SNoLt_trichotomy_or_impred with x1, x2, x1 = x2 leaving 5 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Assume H5: SNoLt x1 x2.
Apply FalseE with x1 = x2.
Apply SNoLt_trichotomy_or_impred with x0, 0, False leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Assume H6: SNoLt x0 0.
Apply SNoLt_irref with mul_SNo x0 x1.
Apply H4 with λ x3 x4 . SNoLt x4 (mul_SNo x0 x1).
Apply neg_mul_SNo_Lt with x0, x2, x1 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
Assume H6: SNoLt 0 x0.
Apply SNoLt_irref with mul_SNo x0 x1.
Apply H4 with λ x3 x4 . SNoLt (mul_SNo x0 x1) x4.
Apply pos_mul_SNo_Lt with x0, x1, x2 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
Assume H5: x1 = x2.
The subproof is completed by applying H5.
Assume H5: SNoLt x2 x1.
Apply FalseE with x1 = x2.
Apply SNoLt_trichotomy_or_impred with x0, 0, False leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Assume H6: SNoLt x0 0.
Apply SNoLt_irref with mul_SNo x0 x1.
Apply H4 with λ x3 x4 . SNoLt (mul_SNo x0 x1) x4.
Apply neg_mul_SNo_Lt with x0, x1, x2 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
Assume H6: SNoLt 0 x0.
Apply SNoLt_irref with mul_SNo x0 x1.
Apply H4 with λ x3 x4 . SNoLt x4 (mul_SNo x0 x1).
Apply pos_mul_SNo_Lt with x0, x2, x1 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
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