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 SNoR : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const SNo_max_of : set set prop const Repl : set (set set) set const minus_SNo : set set term - = minus_SNo axiom minus_SNo_min_max: !x:set.!y:set.(!z:set.z iIn x -> SNo z) -> SNo_min_of x y -> SNo_max_of (Repl x minus_SNo) - y const SNoL : set set axiom SNoL_minus_SNoR: !x:set.SNo x -> SNoL - x = Repl (SNoR x) minus_SNo const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.SNo x -> SNo_min_of (SNoR x) y -> SNo y -> SNo z -> z < x -> (x + x) < y + z -> SNo - x -> SNo - y -> SNo - z -> SNo (x + x) -> SNo (y + z) -> SNo_max_of (SNoL - x) - y -> ?w:set.w iIn SNoL z & y + w = x + x var x:set var y:set var z:set hyp SNo x hyp SNo_min_of (SNoR x) y hyp SNo y hyp SNo z hyp z < x hyp (x + x) < y + z hyp SNo - x hyp SNo - y hyp SNo - z hyp SNo (x + x) claim SNo (y + z) -> ?w:set.w iIn SNoL z & y + w = x + x