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 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const nat_p : set prop const ordsucc : set set axiom nat_ordsucc: !x:set.nat_p x -> nat_p (ordsucc x) axiom nat_trans: !x:set.nat_p x -> !y:set.y iIn x -> Subq y x lemma !x:set.!y:set.!z:set.nat_p x -> y iIn ordsucc (ordsucc z) -> ordsucc z iIn ordsucc x -> Subq (ordsucc z) (ordsucc x) -> y iIn ordsucc x var x:set var y:set var z:set hyp nat_p x hyp z iIn x hyp y iIn ordsucc (ordsucc z) claim ordsucc z iIn ordsucc x -> y iIn ordsucc x