const SNo : set prop 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoCut : set set set const binunion : set set set const Repl : set (set set) set const SNoL : set set const SNoR : set set axiom add_SNo_eq: !x:set.SNo x -> !y:set.SNo y -> x + y = SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set const SNoCutP : set set prop const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo z -> (!v:set.v iIn SNoS_ (SNoLev x) -> (v + y) * z = v * z + y * z) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (x + v) * z = x * z + v * z) -> (!v:set.v iIn SNoS_ (SNoLev z) -> (x + y) * v = x * v + y * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (v + y) * x2 = v * x2 + y * x2) -> (!v:set.v iIn SNoS_ (SNoLev y) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + v) * x2 = x * x2 + v * x2) -> SNoCutP w u -> (!v:set.v iIn w -> !P:prop.(!x2:set.x2 iIn SNoL (x + y) -> !y2:set.y2 iIn SNoL z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> (!x2:set.x2 iIn SNoR (x + y) -> !y2:set.y2 iIn SNoR z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> P) -> (!v:set.v iIn u -> !P:prop.(!x2:set.x2 iIn SNoL (x + y) -> !y2:set.y2 iIn SNoR z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> (!x2:set.x2 iIn SNoR (x + y) -> !y2:set.y2 iIn SNoL z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> P) -> (x + y) * z = SNoCut w u -> SNo (x + y) -> SNo ((x + y) * z) -> SNo (x * z) -> SNo (y * z) -> SNo (x * z + y * z) -> x * z + y * z = SNoCut (binunion (Repl (SNoL (x * z)) \v:set.v + y * z) (Repl (SNoL (y * z)) (add_SNo (x * z)))) (binunion (Repl (SNoR (x * z)) \v:set.v + y * z) (Repl (SNoR (y * z)) (add_SNo (x * z)))) -> (x + y) * z = x * z + y * z var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo z hyp !v:set.v iIn SNoS_ (SNoLev x) -> (v + y) * z = v * z + y * z hyp !v:set.v iIn SNoS_ (SNoLev y) -> (x + v) * z = x * z + v * z hyp !v:set.v iIn SNoS_ (SNoLev z) -> (x + y) * v = x * v + y * v hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (v + y) * x2 = v * x2 + y * x2 hyp !v:set.v iIn SNoS_ (SNoLev y) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + v) * x2 = x * x2 + v * x2 hyp SNoCutP w u hyp !v:set.v iIn w -> !P:prop.(!x2:set.x2 iIn SNoL (x + y) -> !y2:set.y2 iIn SNoL z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> (!x2:set.x2 iIn SNoR (x + y) -> !y2:set.y2 iIn SNoR z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> P hyp !v:set.v iIn u -> !P:prop.(!x2:set.x2 iIn SNoL (x + y) -> !y2:set.y2 iIn SNoR z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> (!x2:set.x2 iIn SNoR (x + y) -> !y2:set.y2 iIn SNoL z -> v = x2 * z + (x + y) * y2 + - x2 * y2 -> P) -> P hyp (x + y) * z = SNoCut w u hyp SNo (x + y) hyp SNo ((x + y) * z) hyp SNo (x * z) hyp SNo (y * z) claim SNo (x * z + y * z) -> (x + y) * z = x * z + y * z