const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const minus_SNo : set set term - = minus_SNo axiom minus_SNo_Lt_contra: !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x const add_SNo : set set set term + = add_SNo infix + 2281 2280 const Empty : set lemma !x:set.!y:set.!z:set.SNo x -> SNo - x -> y = - x + z -> SNo z -> z < x -> SNo - z -> - z + z = Empty -> - x < - z -> y < Empty const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLev : set set const SNoS_ : set set var x:set var y:set var z:set hyp SNo x hyp !w:set.w iIn SNoS_ (SNoLev x) -> - w + w = Empty hyp SNo - x hyp y = - x + z hyp SNo z hyp SNoLev z iIn SNoLev x hyp z < x hyp SNo - z claim - z + z = Empty -> y < Empty