const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const Empty : set const abs_SNo : set set axiom nonneg_abs_SNo: !x:set.Empty <= x -> abs_SNo x = x const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < x + z -> Empty <= y + - x -> abs_SNo (y + - x) = y + - x -> abs_SNo (y + - x) < z var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp x <= y hyp y < x + z claim Empty <= y + - x -> abs_SNo (y + - x) < z