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 In : set set prop term iIn = In infix iIn 2000 2000 const SNoLev : set set const SNoL : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoR : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 var x:set var y:set var z:set var w:set hyp SNo y hyp SNo w hyp w < y hyp SNo (x * y) hyp SNo (z * y) hyp SNo (x * w) hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoR w -> (x * w + u * v) < u * w + x * v hyp SNo (z * w) hyp z iIn SNoL x hyp SNoLev y iIn SNoLev w claim y iIn SNoR w -> (z * y + x * w) < x * y + z * w