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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: SNoLe x0 x1.
Assume H4: SNoLt x1 (add_SNo x0 x2).
Apply nonneg_abs_SNo with add_SNo x1 (minus_SNo x0), λ x3 x4 . SNoLt x4 x2 leaving 2 subgoals.
Apply add_SNo_minus_SNo_rinv with x0, λ x3 x4 . SNoLe x3 (add_SNo x1 (minus_SNo x0)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_Le1 with x0, minus_SNo x0, x1 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply add_SNo_minus_Lt1b with x1, x0, x2 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply add_SNo_com with x2, x0, λ x3 x4 . SNoLt x1 x4 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
The subproof is completed by applying H4.