const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const nat_p : set prop axiom nat_trans: !x:set.nat_p x -> !y:set.y iIn x -> Subq y x const Inj1 : set set const ordsucc : set set lemma !x:set.!y:set.!z:set.nat_p x -> (!w:set.w iIn x -> Inj1 w = ordsucc w) -> y iIn x -> z iIn y -> z iIn x -> ordsucc z iIn Inj1 x var x:set var y:set var z:set hyp nat_p x hyp !w:set.w iIn x -> Inj1 w = ordsucc w hyp y iIn x hyp y = ordsucc z claim z iIn y -> ordsucc z iIn Inj1 x