const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoCut : set set set lemma !x:set.!y:set.SNoCutP x y -> (!z:set.z iIn x -> - - z = z) -> (!z:set.z iIn y -> - - z = z) -> (!z:set.z iIn x -> SNo z) -> (!z:set.z iIn y -> SNo z) -> SNo (SNoCut x y) -> SNo - SNoCut x y -> SNo - - SNoCut x y -> - - SNoCut x y = SNoCut x y var x:set var y:set hyp SNoCutP x y hyp !z:set.z iIn x -> - - z = z hyp !z:set.z iIn y -> - - z = z hyp !z:set.z iIn x -> SNo z hyp !z:set.z iIn y -> SNo z hyp SNo (SNoCut x y) claim SNo - SNoCut x y -> - - SNoCut x y = SNoCut x y