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 SNoS_ : set set const SNoLev : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set const SNoR : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 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 -> SNo y -> SNo z -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (x + x2) * z = x * z + x2 * z) -> (!x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + y) * x2 = x * x2 + y * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x + x2) * y2 = x * y2 + x2 * y2) -> SNo (x + y) -> SNo ((x + y) * z) -> SNo (x * z) -> SNo w -> u iIn SNoL y -> v iIn SNoR z -> (u * z + y * v) <= w + u * v -> SNo u -> u < y -> SNo v -> z < v -> SNo (u * v) -> SNo (x + u) -> SNo (w + u * v) -> SNo ((x + y) * v) -> SNo (u * z) -> SNo (x * v) -> SNo (y * v) -> (x + y) * z < w + x * z var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp SNo z hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> (x + x2) * z = x * z + x2 * z hyp !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + y) * x2 = x * x2 + y * x2 hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x + x2) * y2 = x * y2 + x2 * y2 hyp SNo (x + y) hyp SNo ((x + y) * z) hyp SNo (x * z) hyp SNo w hyp u iIn SNoL y hyp v iIn SNoR z hyp (u * z + y * v) <= w + u * v hyp SNo u hyp u < y hyp SNo v hyp z < v hyp SNo (u * v) hyp SNo (x + u) hyp SNo (w + u * v) hyp SNo ((x + y) * v) hyp SNo (u * z) claim SNo (x * v) -> (x + y) * z < w + x * z