const In : set set prop term iIn = In infix iIn 2000 2000 const SNoR : set set 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 minus_SNo : set set term - = minus_SNo const SNoS_ : set set const SNoLev : set set const SNo : set prop const SNoCut : set set set var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> x2 * y = y * x2 hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> x * x2 = x2 * x hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> x2 * y2 = y2 * x2 hyp !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 hyp !x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn w hyp !x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> x2 * y + x * y2 + - x2 * y2 iIn w hyp !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 hyp !x2:set.x2 iIn SNoL y -> !y2:set.y2 iIn SNoR x -> x2 * x + y * y2 + - x2 * y2 iIn v hyp !x2:set.x2 iIn SNoR y -> !y2:set.y2 iIn SNoL x -> x2 * x + y * y2 + - x2 * y2 iIn v hyp z = u claim w = v -> SNoCut z w = SNoCut u v