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 x -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.SNo x2 -> !P:prop.(SNo (v * x2) -> (!y2:set.y2 iIn SNoL v -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + v * z2) < v * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR v -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + v * z2) < v * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL v -> !z2:set.z2 iIn SNoR x2 -> (v * x2 + y2 * z2) < y2 * x2 + v * z2) -> (!y2:set.y2 iIn SNoR v -> !z2:set.z2 iIn SNoL x2 -> (v * x2 + y2 * z2) < y2 * x2 + v * z2) -> P) -> P) -> SNo y -> (!v:set.v iIn SNoR x -> !x2:set.SNo x2 -> SNo (v * x2)) -> z iIn SNoL x -> SNo z -> SNoLev z iIn SNoLev x -> w iIn SNoR x -> SNo w -> SNo (z * y) -> SNo (w * y) -> z < w -> u iIn SNoL y -> SNo u -> SNo (z * u) -> SNo (w * u) -> (z * y + w * u) < w * y + z * u var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.SNo x2 -> !P:prop.(SNo (v * x2) -> (!y2:set.y2 iIn SNoL v -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + v * z2) < v * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR v -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + v * z2) < v * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL v -> !z2:set.z2 iIn SNoR x2 -> (v * x2 + y2 * z2) < y2 * x2 + v * z2) -> (!y2:set.y2 iIn SNoR v -> !z2:set.z2 iIn SNoL x2 -> (v * x2 + y2 * z2) < y2 * x2 + v * z2) -> P) -> P hyp SNo y hyp !v:set.v iIn SNoL x -> !x2:set.SNo x2 -> SNo (v * x2) hyp !v:set.v iIn SNoR x -> !x2:set.SNo x2 -> SNo (v * x2) hyp z iIn SNoL x hyp SNo z hyp SNoLev z iIn SNoLev x hyp w iIn SNoR x hyp SNo w hyp SNo (z * y) hyp SNo (w * y) hyp z < w hyp u iIn SNoL y hyp SNo u claim SNo (z * u) -> (z * y + w * u) < w * y + z * u