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