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