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 const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.(!z2:set.z2 iIn SNoS_ (SNoLev x) -> !w2:set.SNo w2 -> !P:prop.(SNo (z2 * w2) -> (!u2:set.u2 iIn SNoL z2 -> !v2:set.v2 iIn SNoL w2 -> (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2) -> (!u2:set.u2 iIn SNoR z2 -> !v2:set.v2 iIn SNoR w2 -> (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2) -> (!u2:set.u2 iIn SNoL z2 -> !v2:set.v2 iIn SNoR w2 -> (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2) -> (!u2:set.u2 iIn SNoR z2 -> !v2:set.v2 iIn SNoL w2 -> (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2) -> P) -> P) -> SNo y -> (!z2:set.z2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * z2) -> (!w2:set.w2 iIn SNoL x -> !u2:set.u2 iIn SNoL z2 -> (w2 * z2 + x * u2) < x * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoR x -> !u2:set.u2 iIn SNoR z2 -> (w2 * z2 + x * u2) < x * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoL x -> !u2:set.u2 iIn SNoR z2 -> (x * z2 + w2 * u2) < w2 * z2 + x * u2) -> (!w2:set.w2 iIn SNoR x -> !u2:set.u2 iIn SNoL z2 -> (x * z2 + w2 * u2) < w2 * z2 + x * u2) -> P) -> P) -> u iIn SNoS_ (SNoLev x) -> v iIn SNoS_ (SNoLev x) -> x2 iIn SNoS_ (SNoLev y) -> y2 iIn SNoS_ (SNoLev y) -> SNo x2 -> SNo y2 -> SNo (u * x2) -> SNo (v * y2) -> SNo (u * y) -> SNo (x * x2) -> !P:prop.(SNo (u * y) -> SNo (x * x2) -> SNo (u * x2) -> SNo (v * y) -> SNo (x * y2) -> SNo (v * y2) -> SNo (u * y2) -> SNo (v * x2) -> (z = u * y + x * x2 + - u * x2 -> w = v * y + x * y2 + - v * y2 -> (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 -> z < w) -> P) -> P var x:set var y:set var z:set var w:set var u:set var v:set var x2:set var y2:set hyp !z2:set.z2 iIn SNoS_ (SNoLev x) -> !w2:set.SNo w2 -> !P:prop.(SNo (z2 * w2) -> (!u2:set.u2 iIn SNoL z2 -> !v2:set.v2 iIn SNoL w2 -> (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2) -> (!u2:set.u2 iIn SNoR z2 -> !v2:set.v2 iIn SNoR w2 -> (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2) -> (!u2:set.u2 iIn SNoL z2 -> !v2:set.v2 iIn SNoR w2 -> (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2) -> (!u2:set.u2 iIn SNoR z2 -> !v2:set.v2 iIn SNoL w2 -> (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2) -> P) -> P hyp SNo y hyp !z2:set.z2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * z2) -> (!w2:set.w2 iIn SNoL x -> !u2:set.u2 iIn SNoL z2 -> (w2 * z2 + x * u2) < x * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoR x -> !u2:set.u2 iIn SNoR z2 -> (w2 * z2 + x * u2) < x * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoL x -> !u2:set.u2 iIn SNoR z2 -> (x * z2 + w2 * u2) < w2 * z2 + x * u2) -> (!w2:set.w2 iIn SNoR x -> !u2:set.u2 iIn SNoL z2 -> (x * z2 + w2 * u2) < w2 * z2 + x * u2) -> P) -> P hyp u iIn SNoS_ (SNoLev x) hyp v iIn SNoS_ (SNoLev x) hyp x2 iIn SNoS_ (SNoLev y) hyp y2 iIn SNoS_ (SNoLev y) hyp SNo x2 hyp SNo y2 hyp SNo (u * x2) hyp SNo (v * y2) claim SNo (u * y) -> !P:prop.(SNo (u * y) -> SNo (x * x2) -> SNo (u * x2) -> SNo (v * y) -> SNo (x * y2) -> SNo (v * y2) -> SNo (u * y2) -> SNo (v * x2) -> (z = u * y + x * x2 + - u * x2 -> w = v * y + x * y2 + - v * y2 -> (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 -> z < w) -> P) -> P