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 In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoR : set set axiom mul_SNo_prop_1: !x:set.SNo x -> !y:set.SNo y -> !P:prop.(SNo (x * y) -> (!z:set.z iIn SNoL x -> !w:set.w iIn SNoL y -> (z * y + x * w) < x * y + z * w) -> (!z:set.z iIn SNoR x -> !w:set.w iIn SNoR y -> (z * y + x * w) < x * y + z * w) -> (!z:set.z iIn SNoL x -> !w:set.w iIn SNoR y -> (x * y + z * w) < z * y + x * w) -> (!z:set.z iIn SNoR x -> !w:set.w iIn SNoL y -> (x * y + z * w) < z * y + x * w) -> P) -> P lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z < x -> w < y -> SNo (x * y) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> (u * y + x * v) < x * y + u * v) -> SNo (z * y) -> (!u:set.u iIn SNoR z -> !v:set.v iIn SNoL y -> (z * y + u * v) < u * y + z * v) -> SNo (x * w) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR w -> (x * w + u * v) < u * w + x * v) -> SNo (z * w) -> (!u:set.u iIn SNoR z -> !v:set.v iIn SNoR w -> (u * w + z * v) < z * w + u * v) -> SNo (z * y + x * w) -> (z * y + x * w) < x * y + z * w claim !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