const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoL : set set axiom SNoL_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> y < x -> y iIn SNoL x const binintersect : set set set axiom binintersectE: !x:set.!y:set.!z:set.z iIn binintersect x y -> z iIn x & z iIn y const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x const SNoEq_ : set set set prop const nIn : set set prop axiom SNoLtE: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.SNo z -> SNoLev z iIn binintersect (SNoLev x) (SNoLev y) -> SNoEq_ (SNoLev z) z x -> SNoEq_ (SNoLev z) z y -> x < z -> z < y -> nIn (SNoLev z) x -> SNoLev z iIn y -> P) -> (SNoLev x iIn SNoLev y -> SNoEq_ (SNoLev x) x y -> SNoLev x iIn y -> P) -> (SNoLev y iIn SNoLev x -> SNoEq_ (SNoLev y) x y -> nIn (SNoLev y) x -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo (x * y) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> (v * y + x * x2) < x * y + v * x2) -> SNo (z * y) -> SNo (x * w) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR w -> (x * w + v * x2) < v * w + x * x2) -> SNo (z * w) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> z iIn SNoL x -> SNo u -> w < u -> u < y -> SNoLev u iIn SNoLev w -> SNoLev u iIn SNoLev y -> u iIn SNoL y -> (z * y + x * w) < x * y + z * w lemma !x:set.!y:set.!z:set.!w:set.SNo y -> SNo w -> w < y -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> (u * y + x * v) < x * y + u * v) -> z iIn SNoL x -> SNoLev w iIn SNoLev y -> w iIn SNoL y -> (z * y + x * w) < x * y + z * w lemma !x:set.!y:set.!z:set.!w:set.SNo y -> SNo w -> w < y -> SNo (x * y) -> SNo (z * y) -> 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) -> z iIn SNoL x -> SNoLev y iIn SNoLev w -> y iIn SNoR w -> (z * y + x * w) < x * y + z * w var x:set var y:set var z:set var w:set hyp SNo x hyp SNo y hyp SNo z hyp SNo w hyp z < x hyp w < y hyp SNo (x * y) hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> (u * y + x * v) < x * y + u * v hyp SNo (z * y) hyp SNo (x * w) hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoR w -> (x * w + u * v) < u * w + x * v hyp SNo (z * w) hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp SNoLev z iIn SNoLev x claim z iIn SNoL x -> (z * y + x * w) < x * y + z * w