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 add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_Lt_subprop2: !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> SNo v -> (x + u) < z + v -> (y + v) < w + u -> (x + y) < z + w axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const SNoR : set set const SNoS_ : set set const SNoLev : set set var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.SNo y2 -> !P:prop.(SNo (x2 * y2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoR y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoL y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> P) -> P hyp SNo y hyp w iIn SNoR x hyp SNo (z * y) hyp SNo (w * y) hyp u iIn SNoL y hyp SNo (z * u) hyp SNo (w * u) hyp v iIn SNoL w hyp SNo (v * u) hyp SNo (v * y) hyp (z * y + v * u) < z * u + v * y claim (w * u + v * y) < w * y + v * u -> (z * y + w * u) < w * y + z * u