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 omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom add_SNo_Lt1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < z -> (x + y) < z + y axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const Empty : set const ordsucc : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 var x:set var y:set var z:set var w:set hyp SNo x hyp y iIn omega hyp x < z + eps_ y hyp SNo z hyp z <= Empty hyp SNo (eps_ (ordsucc w)) hyp Empty < eps_ (ordsucc w) claim z < eps_ (ordsucc w) -> x < eps_ (ordsucc w) + eps_ y