const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoCutP : set set prop const SNoCut : set set set const Repl : set (set set) set axiom minus_SNoCut_eq: !x:set.!y:set.SNoCutP x y -> - SNoCut x y = SNoCut (Repl y minus_SNo) (Repl x minus_SNo) const SNoL : set set 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 SNoR : set set axiom mul_SNo_eq_3: !x:set.!y:set.SNo x -> SNo y -> !P:prop.(!z:set.!w:set.SNoCutP z w -> (!u:set.u iIn z -> !Q:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> Q) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> Q) -> Q) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn z) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn z) -> (!u:set.u iIn w -> !Q:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> Q) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> Q) -> Q) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn w) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn w) -> x * y = SNoCut z w -> P) -> P const SNoS_ : set set const SNoLev : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2) -> SNo - x -> SNoCutP z w -> (!x2:set.x2 iIn z -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> x2 * y + x * y2 + - x2 * y2 iIn z) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn z) -> (!x2:set.x2 iIn w -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn w) -> (!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 u -> !P:prop.(!y2:set.y2 iIn SNoL - x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR - x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoL y -> x2 * y + (- x) * y2 + - x2 * y2 iIn u) -> (!x2:set.x2 iIn SNoR - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn u) -> (!x2:set.x2 iIn v -> !P:prop.(!y2:set.y2 iIn SNoL - x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR - x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn v) -> (!x2:set.x2 iIn SNoR - x -> !y2:set.y2 iIn SNoL y -> x2 * y + (- x) * y2 + - x2 * y2 iIn v) -> (- x) * y = SNoCut u v -> - x * y = SNoCut (Repl w minus_SNo) (Repl z minus_SNo) -> (- x) * y = - x * y var x:set var y:set hyp SNo x hyp SNo y hyp !z:set.z iIn SNoS_ (SNoLev x) -> (- z) * y = - z * y hyp !z:set.z iIn SNoS_ (SNoLev y) -> (- x) * z = - x * z hyp !z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> (- z) * w = - z * w claim SNo - x -> (- x) * y = - x * y