const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_min_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> y <= z 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 SNoL : set set const minus_SNo : set set term - = minus_SNo const SNoR : set set axiom SNoL_minus_SNoR: !x:set.SNo x -> SNoL - x = Repl (SNoR x) minus_SNo axiom minus_SNo_invol: !x:set.SNo x -> - - x = x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom minus_add_SNo_distr: !x:set.!y:set.SNo x -> SNo y -> - (x + y) = - x + - y var x:set var y:set var z:set var w:set hyp SNo y hyp SNo z hyp SNo - y hyp SNo - z hyp SNo (x + x) hyp w iIn SNoR - z hyp - y + w = - (x + x) hyp SNo w claim SNo - w -> ?u:set.u iIn SNoL z & y + u = x + x