const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 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 axiom add_SNo_Lt_subprop2: !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) < z + v -> (y + v) < w + u -> (x + y) < z + w lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> SNo v -> SNo x2 -> SNo y2 -> (x + v) < x2 + y2 -> (y + y2) < u -> (x2 + z) < w + v -> (x + z) < y2 + w -> (x + y + z) < w + u claim !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> SNo v -> SNo x2 -> SNo y2 -> (x + v) < x2 + y2 -> (y + y2) < u -> (x2 + z) < w + v -> (x + y + z) < w + u