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 SNoCut : set set set axiom SNoCutP_SNoCut_L: !x:set.!y:set.SNoCutP x y -> !z:set.z iIn x -> z < SNoCut x y const SNoL : set set const SNoLev : set set axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const SNoR : set set axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P axiom SNoCutP_SNoCut_R: !x:set.!y:set.SNoCutP x y -> !z:set.z iIn y -> SNoCut x y < z 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 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo y -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> x2 * y + x * y2 + - x2 * y2 iIn z) -> x * y = SNoCut z w -> (!x2:set.x2 iIn SNoL x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoL y -> SNo (x * x2)) -> SNoCutP z w -> SNo (x * y) -> u iIn SNoL x -> v iIn SNoL y -> SNo v -> (u * y + x * v + - u * v) < x * y -> (u * y + x * v) < x * y + u * v lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo y -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn z) -> x * y = SNoCut z w -> (!x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoR y -> SNo (x * x2)) -> SNoCutP z w -> SNo (x * y) -> u iIn SNoR x -> v iIn SNoR y -> SNo v -> (u * y + x * v + - u * v) < x * y -> (u * y + x * v) < x * y + u * v lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo y -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn w) -> x * y = SNoCut z w -> (!x2:set.x2 iIn SNoL x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoR y -> SNo (x * x2)) -> SNoCutP z w -> SNo (x * y) -> u iIn SNoL x -> v iIn SNoR y -> SNo v -> x * y < u * y + x * v + - u * v -> (x * y + u * v) < u * y + x * v lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo y -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> x2 * y + x * y2 + - x2 * y2 iIn w) -> x * y = SNoCut z w -> (!x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoL y -> SNo (x * x2)) -> SNoCutP z w -> SNo (x * y) -> u iIn SNoR x -> v iIn SNoL y -> SNo v -> x * y < u * y + x * v + - u * v -> (x * y + u * v) < u * y + x * v var x:set var y:set var z:set var w:set hyp SNo y hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn z hyp !u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn z hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn w hyp !u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn w hyp x * y = SNoCut z w hyp !u:set.u iIn SNoL x -> !v:set.SNo v -> SNo (u * v) hyp !u:set.u iIn SNoR x -> !v:set.SNo v -> SNo (u * v) hyp !u:set.u iIn SNoL y -> SNo (x * u) hyp !u:set.u iIn SNoR y -> SNo (x * u) hyp SNoCutP z w claim SNo (x * y) -> !P:prop.(SNo (x * y) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> (u * y + x * v) < x * y + u * v) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> (u * y + x * v) < x * y + u * v) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> (x * y + u * v) < u * y + x * v) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> (x * y + u * v) < u * y + x * v) -> P) -> P