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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 (add_SNo x0 x1) (add_SNo x2 x1).
Apply add_SNo_minus_R2 with x0, x1, λ x3 x4 . SNoLe x3 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_minus_R2 with x2, x1, λ x3 x4 . SNoLe (add_SNo (add_SNo x0 x1) (minus_SNo x1)) x3 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply add_SNo_Le1 with add_SNo x0 x1, minus_SNo x1, add_SNo x2 x1 leaving 4 subgoals.
Apply SNo_add_SNo with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with x1.
The subproof is completed by applying H1.
Apply SNo_add_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 H3.