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 SNo : set prop 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 SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y <= z -> (x + y) <= x + z const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom add_SNo_Lt1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < z -> (x + y) < z + y axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z const SNoL : set set const SNoS_ : set set const SNoLev : set set var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo z hyp !v:set.v iIn SNoS_ (SNoLev y) -> x + v + z = (x + v) + z hyp SNo (x + y) hyp SNo w hyp u iIn SNoL y hyp w <= u + z hyp SNo u hyp u < y claim x + u + z = (x + u) + z -> (x + w) < (x + y) + z