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 term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term nIn = \x:set.\y:set.~ x iIn y const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_ordinal_ordinal: !x:set.ordinal x -> !y:set.ordinal y -> ordinal (x + y) const ordsucc : set set const SNo : set prop lemma !x:set.!y:set.ordinal x -> ordinal y -> (!z:set.z iIn y -> ordsucc x + z = ordsucc (x + z)) -> SNo x -> SNo y -> ordinal (x + y) -> ordinal (ordsucc x) -> SNo (ordsucc x) -> ordinal (ordsucc x + y) -> ordsucc x + y = ordsucc (x + y) var x:set var y:set hyp ordinal x hyp ordinal y hyp !z:set.z iIn y -> ordsucc x + z = ordsucc (x + z) hyp SNo x hyp SNo y hyp ordinal (x + y) hyp ordinal (ordsucc x) claim SNo (ordsucc x) -> ordsucc x + y = ordsucc (x + y)