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