const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoS_ : set set const SNoLev : set set axiom SNoL_SNoS: !x:set.SNo x -> !y:set.y iIn SNoL x -> y iIn SNoS_ (SNoLev x) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) 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 add_SNo_com_3_0_1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = y + x + z const SNoLt : set set prop term < = SNoLt infix < 2020 2020 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_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) 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 const Subq : set set prop axiom mul_SNo_Subq_lem: !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.(!z2:set.z2 iIn x2 -> !P:prop.(!w2:set.w2 iIn z -> !u2:set.u2 iIn w -> z2 = w2 * y + x * u2 + - w2 * u2 -> P) -> (!w2:set.w2 iIn u -> !u2:set.u2 iIn v -> z2 = w2 * y + x * u2 + - w2 * u2 -> P) -> P) -> (!z2:set.z2 iIn z -> !w2:set.w2 iIn w -> z2 * y + x * w2 + - z2 * w2 iIn y2) -> (!z2:set.z2 iIn u -> !w2:set.w2 iIn v -> z2 * y + x * w2 + - z2 * w2 iIn y2) -> Subq x2 y2 axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y const SNoCutP : set set prop const SNoCut : set 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 axiom SNoLev_ind2: !r:set set prop.(!x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> r z y) -> (!z:set.z iIn SNoS_ (SNoLev y) -> r x z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> r z w) -> r x y) -> !x:set.!y:set.SNo x -> SNo y -> r x y 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 = y * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> x * x2 = x2 * x) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> x2 * y2 = y2 * x2) -> (!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) -> (!x2:set.x2 iIn u -> !P:prop.(!y2:set.y2 iIn SNoL y -> !z2:set.z2 iIn SNoL x -> x2 = y2 * x + y * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR y -> !z2:set.z2 iIn SNoR x -> x2 = y2 * x + y * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL y -> !y2:set.y2 iIn SNoL x -> x2 * x + y * y2 + - x2 * y2 iIn u) -> (!x2:set.x2 iIn SNoR y -> !y2:set.y2 iIn SNoR x -> x2 * x + y * y2 + - x2 * y2 iIn u) -> (!x2:set.x2 iIn v -> !P:prop.(!y2:set.y2 iIn SNoL y -> !z2:set.z2 iIn SNoR x -> x2 = y2 * x + y * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR y -> !z2:set.z2 iIn SNoL x -> x2 = y2 * x + y * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL y -> !y2:set.y2 iIn SNoR x -> x2 * x + y * y2 + - x2 * y2 iIn v) -> (!x2:set.x2 iIn SNoR y -> !y2:set.y2 iIn SNoL x -> x2 * x + y * y2 + - x2 * y2 iIn v) -> z = u -> SNoCut z w = SNoCut u v claim !x:set.!y:set.SNo x -> SNo y -> x * y = y * x