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 Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f const minus_SNo : set set term - = minus_SNo const SNoL : 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 SNoS_ : set set const SNoLev : set set var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp !v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y hyp !v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2 hyp !v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> v * y + x * x2 + - v * x2 iIn z hyp u iIn SNoL y hyp SNo w hyp SNo u hyp SNo - w hyp - w iIn SNoL x claim w * y + (- x) * u + - w * u = - ((- w) * y + x * u + - (- w) * u) -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo