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.!v:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo (x * y) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> (x2 * y + x * y2) < x * y + x2 * y2) -> SNo (z * y) -> (!x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoL y -> (z * y + x2 * y2) < x2 * y + z * y2) -> SNo (x * w) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR w -> (x * w + x2 * y2) < x2 * w + x * y2) -> SNo (z * w) -> (!x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoR w -> (x2 * w + z * y2) < z * w + x2 * y2) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> SNo u -> u iIn SNoL x -> u iIn SNoR z -> SNo (u * y) -> SNo (u * w) -> SNo v -> w < v -> v < y -> SNoLev v iIn SNoLev w -> SNoLev v iIn SNoLev y -> v iIn SNoL y -> (z * y + x * w) < x * y + z * w lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo y -> SNo w -> w < y -> 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) -> (!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) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> u iIn SNoL x -> u iIn SNoR z -> SNo (u * y) -> SNo (u * w) -> SNo (u * y + x * w) -> SNo (u * y + z * w) -> SNo (x * y + u * w) -> SNo (z * y + u * w) -> 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.!u:set.SNo y -> SNo w -> w < y -> SNo (x * y) -> 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) -> (!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) -> u iIn SNoL x -> u iIn SNoR z -> SNo (u * y) -> SNo (u * w) -> SNo (z * w + u * y) -> SNo (u * w + x * y) -> SNo (x * w + u * y) -> SNo (u * w + z * y) -> 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 var u:set hyp SNo x hyp SNo y hyp SNo z hyp SNo w hyp w < y hyp SNo (x * y) hyp !v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> (v * y + x * x2) < x * y + v * x2 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 !v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR w -> (x * w + v * x2) < v * w + x * x2 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 SNo u hyp u iIn SNoL x hyp u iIn SNoR z hyp SNo (u * y) hyp SNo (u * w) hyp SNo (u * y + x * w) hyp SNo (u * y + z * w) hyp SNo (x * y + u * w) hyp SNo (z * y + u * w) hyp SNo (z * w + u * y) hyp SNo (u * w + x * y) hyp SNo (x * w + u * y) claim SNo (u * w + z * y) -> (z * y + x * w) < x * y + z * w