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 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 const SNoL : set set const SNoS_ : set set const SNoLev : set set var x:set var y:set var z:set var w:set hyp SNo x hyp SNo y hyp SNo z hyp !u:set.u iIn SNoS_ (SNoLev z) -> x + y + u = (x + y) + u hyp SNo (y + z) hyp w iIn SNoL z hyp SNo w hyp w < z claim x + y + w = (x + y) + w -> ((x + y) + w) < x + y + z