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) -> SNo (u * y + z * v) -> SNo (u * w + x * v) -> SNo (z * y + u * v) -> SNo (x * w + u * v) -> SNo (u * w + z * v) -> SNo (u * y + x * v) -> SNo (x * y + u * v) -> SNo (z * w + u * v) -> ((z * y + x * w) + u * v + u * v) < (x * y + z * w) + u * v + u * v 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 SNo (x * w) hyp SNo (z * w) hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp SNo (u * y) hyp SNo (u * w) 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 hyp (z * y + u * v) < u * y + z * v hyp SNo (u * v + u * v) hyp SNo (u * y + z * v) hyp SNo (u * w + x * v) hyp SNo (z * y + u * v) hyp SNo (x * w + u * v) hyp SNo (u * w + z * v) hyp SNo (u * y + x * v) claim SNo (x * y + u * v) -> ((z * y + x * w) + u * v + u * v) < (x * y + z * w) + u * v + u * v