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 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const minus_SNo : set set term - = minus_SNo var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo (z * u) hyp SNo (w * v) hyp SNo (w * y) hyp SNo (x * v) hyp (z * y + x * u + w * v) < w * y + x * v + z * u hyp (z * y + x * u + - z * u) + z * u + w * v = z * y + x * u + w * v claim (w * y + x * v + - w * v) + z * u + w * v = w * y + x * v + z * u -> ((z * y + x * u + - z * u) + z * u + w * v) < (w * y + x * v + - w * v) + z * u + w * v