const SNo : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const Empty : set axiom mul_SNo_zeroL: !x:set.SNo x -> Empty * x = Empty axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_0L: !x:set.SNo x -> Empty + x = x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.!z:set.SNo x -> x < Empty -> SNo y -> SNo z -> z < y -> x * y + Empty * z = x * y -> Empty * y + x * z = x * z -> x * y < x * z var x:set var y:set var z:set hyp SNo x hyp x < Empty hyp SNo y hyp SNo z hyp z < y claim x * y + Empty * z = x * y -> x * y < x * z