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 minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - 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) axiom add_SNo_minus_SNo_prop5: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x + y + - z) + z + w = x + y + w axiom add_SNo_Lt1_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < z + y -> x < z const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo (z * u) -> SNo (w * v) -> SNo (z * y) -> SNo (x * u) -> SNo (w * y) -> SNo (x * v) -> (z * y + x * u + w * v) < w * y + x * v + z * u -> (z * y + x * u + - z * u) + z * u + w * v = z * y + x * u + w * v -> ((z * y + x * u + - z * u) + z * u + w * v) < (w * y + x * v + - w * v) + z * u + w * v const SNoS_ : set set const SNoLev : set set const SNoL : set set const SNoR : set set 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 v iIn SNoS_ (SNoLev x) hyp SNo x2 hyp SNo (u * x2) hyp SNo (v * y2) hyp SNo (u * y) hyp SNo (x * x2) hyp SNo (v * y) hyp SNo (x * y2) hyp SNo (u * y2) claim SNo (v * 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