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 SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom add_SNo_minus_Lt1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + - y) < z -> x < z + y const SNoR : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoCut : set 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 SNoR x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn z hyp x * y = SNoCut z w hyp !x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2) hyp !x2:set.x2 iIn SNoR y -> SNo (x * x2) hyp SNoCutP z w hyp SNo (x * y) hyp u iIn SNoR x hyp v iIn SNoR y hyp SNo v claim (u * y + x * v + - u * v) < x * y -> (u * y + x * v) < x * y + u * v