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 minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo z -> (!v:set.v iIn SNoS_ (SNoLev x) -> (v + y) * z = v * z + y * z) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (x + v) * z = x * z + v * z) -> (!v:set.v iIn SNoS_ (SNoLev z) -> (x + y) * v = x * v + y * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (v + y) * x2 = v * x2 + y * x2) -> (!v:set.v iIn SNoS_ (SNoLev y) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + v) * x2 = x * x2 + v * x2) -> SNo (x + y) -> SNo (x * z) -> SNo (y * z) -> SNo (x * z + y * z) -> w iIn SNoR (x + y) -> u iIn SNoR z -> SNo w -> SNo u -> z < u -> SNo (x * u) -> SNo (y * u) -> SNo (w * z) -> SNo ((x + y) * u) -> (w * z + (x + y) * u + - w * u) < x * z + y * z var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo z hyp !v:set.v iIn SNoS_ (SNoLev x) -> (v + y) * z = v * z + y * z hyp !v:set.v iIn SNoS_ (SNoLev y) -> (x + v) * z = x * z + v * z hyp !v:set.v iIn SNoS_ (SNoLev z) -> (x + y) * v = x * v + y * v hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (v + y) * x2 = v * x2 + y * x2 hyp !v:set.v iIn SNoS_ (SNoLev y) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + v) * x2 = x * x2 + v * x2 hyp SNo (x + y) hyp SNo (x * z) hyp SNo (y * z) hyp SNo (x * z + y * z) hyp w iIn SNoR (x + y) hyp u iIn SNoR z hyp SNo w hyp SNo u hyp z < u hyp SNo (x * u) hyp SNo (y * u) claim SNo (w * z) -> (w * z + (x + y) * u + - w * u) < x * z + y * z