const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x 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 -> (x + u) < v + w -> x + y + z + u = (y + z) + x + u -> (y + z + w) + v = (y + z) + v + w -> (x + y + z + u) < (y + z + w) + v var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp SNo z hyp SNo w hyp SNo u hyp SNo v hyp (x + u) < v + w claim x + y + z + u = (y + z) + x + u -> (x + y + z + u) < (y + z + w) + v