const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop const ordsucc : set set const Empty : set axiom SNo_1: SNo (ordsucc Empty) axiom In_0_1: Empty iIn ordsucc Empty const SNoL : set set axiom SNoL_1: SNoL (ordsucc Empty) = ordsucc Empty const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set axiom mul_SNo_prop_1: !x:set.SNo x -> !y:set.SNo y -> !P:prop.(SNo (x * y) -> (!z:set.z iIn SNoL x -> !w:set.w iIn SNoL y -> (z * y + x * w) < x * y + z * w) -> (!z:set.z iIn SNoR x -> !w:set.w iIn SNoR y -> (z * y + x * w) < x * y + z * w) -> (!z:set.z iIn SNoL x -> !w:set.w iIn SNoR y -> (x * y + z * w) < z * y + x * w) -> (!z:set.z iIn SNoR x -> !w:set.w iIn SNoL y -> (x * y + z * w) < z * y + x * w) -> P) -> P const SNoCutP : set set prop const minus_SNo : set set term - = minus_SNo 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 const SNoS_ : set set const SNoLev : set set axiom SNoLev_ind: !p:set prop.(!x:set.SNo x -> (!y:set.y iIn SNoS_ (SNoLev x) -> p y) -> p x) -> !x:set.SNo x -> p x lemma !x:set.!y:set.!z:set.SNo x -> (!w:set.w iIn SNoS_ (SNoLev x) -> w * ordsucc Empty = w) -> SNoCutP y z -> (!w:set.w iIn y -> !P:prop.(!u:set.u iIn SNoL x -> !v:set.v iIn SNoL (ordsucc Empty) -> w = u * ordsucc Empty + x * v + - u * v -> P) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR (ordsucc Empty) -> w = u * ordsucc Empty + x * v + - u * v -> P) -> P) -> (!w:set.w iIn z -> !P:prop.(!u:set.u iIn SNoL x -> !v:set.v iIn SNoR (ordsucc Empty) -> w = u * ordsucc Empty + x * v + - u * v -> P) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL (ordsucc Empty) -> w = u * ordsucc Empty + x * v + - u * v -> P) -> P) -> x * ordsucc Empty = SNoCut y z -> (!w:set.w iIn SNoL x -> !u:set.u iIn SNoL (ordsucc Empty) -> (w * ordsucc Empty + x * u) < x * ordsucc Empty + w * u) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoL (ordsucc Empty) -> (x * ordsucc Empty + w * u) < w * ordsucc Empty + x * u) -> Empty iIn SNoL (ordsucc Empty) -> x * ordsucc Empty = x claim !x:set.SNo x -> x * ordsucc Empty = x