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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 (minus_SNo x1)) x2.
Apply SNoLeE with add_SNo x0 (minus_SNo x1), x2, SNoLe x0 (add_SNo x2 x1) leaving 5 subgoals.
Apply SNo_add_SNo with x0, minus_SNo x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with x1.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Assume H4: SNoLt (add_SNo x0 (minus_SNo x1)) x2.
Apply SNoLtLe with x0, add_SNo x2 x1.
Apply add_SNo_minus_Lt1 with x0, x1, x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Assume H4: add_SNo x0 (minus_SNo x1) = x2.
Apply H4 with λ x3 x4 . SNoLe x0 (add_SNo x3 x1).
Apply add_SNo_minus_R2' with x0, x1, λ x3 x4 . SNoLe x0 x4 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNoLe_ref with x0.