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