const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const SNoS_ : set set const SNoLev : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo y -> (!v:set.v iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * v) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR v -> (x * v + x2 * y2) < x2 * v + x * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL v -> (x * v + x2 * y2) < x2 * v + x * y2) -> P) -> P) -> (!v:set.v iIn SNoL x -> !x2:set.SNo x2 -> SNo (v * x2)) -> z iIn SNoL y -> SNo z -> SNoLev z iIn SNoLev y -> w iIn SNoR y -> SNo w -> SNo (x * z) -> SNo (x * w) -> z < w -> u iIn SNoL x -> SNo (u * z) -> SNo (u * w) -> (x * z + u * w) < u * z + x * w var x:set var y:set var z:set var w:set var u:set hyp SNo y hyp !v:set.v iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * v) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR v -> (x * v + x2 * y2) < x2 * v + x * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL v -> (x * v + x2 * y2) < x2 * v + x * y2) -> P) -> P hyp !v:set.v iIn SNoL x -> !x2:set.SNo x2 -> SNo (v * x2) hyp z iIn SNoL y hyp SNo z hyp SNoLev z iIn SNoLev y hyp w iIn SNoR y hyp SNo w hyp SNo (x * z) hyp SNo (x * w) hyp z < w hyp u iIn SNoL x claim SNo (u * z) -> (x * z + u * w) < u * z + x * w