const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo (x * y) -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> SNo (u * y) -> SNo (u * w) -> SNo (x * v) -> SNo (z * v) -> SNo (u * v) -> (u * y + x * v) < x * y + u * v -> (u * w + z * v) < z * w + u * v -> (x * w + u * v) < u * w + x * v -> (z * y + u * v) < u * y + z * v -> SNo (u * v + u * v) -> (z * y + x * w) < x * y + z * w const In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoR : set set var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo (x * y) hyp SNo (z * y) hyp !x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoL y -> (z * y + x2 * y2) < x2 * y + z * y2 hyp SNo (x * w) hyp SNo (z * w) hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp u iIn SNoR z hyp SNo (u * y) hyp SNo (u * w) hyp v iIn SNoL y hyp SNo (x * v) hyp SNo (z * v) hyp SNo (u * v) hyp (u * y + x * v) < x * y + u * v hyp (u * w + z * v) < z * w + u * v hyp (x * w + u * v) < u * w + x * v claim (z * y + u * v) < u * y + z * v -> (z * y + x * w) < x * y + z * w