const SNo : set prop const Empty : set axiom SNo_0: SNo Empty const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom mul_SNo_Lt: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z < x -> w < y -> (z * y + x * w) < x * y + z * w axiom mul_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x * y = y * x axiom mul_SNo_zeroR: !x:set.SNo x -> x * Empty = Empty axiom add_SNo_0R: !x:set.SNo x -> x + Empty = x claim !x:set.!y:set.SNo x -> SNo y -> Empty < x -> Empty < y -> Empty < x * y