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 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> SNo v -> (y + v) < u -> (x + z) < v + w -> SNo (x + z) -> (x + y + z) < w + u var x:set var y:set var z:set var w:set var u:set var v:set var x2:set var y2:set hyp SNo x hyp SNo y hyp SNo z hyp SNo w hyp SNo u hyp SNo v hyp SNo x2 hyp SNo y2 hyp (x + v) < x2 + y2 hyp (y + y2) < u hyp (x2 + z) < w + v claim (x + z) < y2 + w -> (x + y + z) < w + u