const SNo : set prop const Empty : set axiom SNo_0: SNo Empty const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 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 var x:set var y:set var z:set hyp SNo x hyp Empty < x hyp SNo y hyp SNo z hyp y < z hyp Empty * z + x * y = x * y claim x * z + Empty * y = x * z -> x * y < x * z