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 minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x 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 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 const SNo_max_of : set set prop const SNoL : set set axiom double_SNo_max_1: !x:set.!y:set.SNo x -> SNo_max_of (SNoL x) y -> !z:set.SNo z -> x < z -> (y + z) < x + x -> ?w:set.w iIn SNoR z & y + w = x + x lemma !x:set.!y:set.!z:set.!w:set.SNo y -> SNo z -> SNo - y -> SNo - z -> SNo (x + x) -> w iIn SNoR - z -> - y + w = - (x + x) -> SNo w -> SNo - w -> ?u:set.u iIn SNoL z & y + u = x + x var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp (x + x) < y + z hyp SNo - x hyp SNo - y hyp SNo - z hyp SNo (x + x) hyp SNo (y + z) hyp SNo_max_of (SNoL - x) - y hyp - x < - z claim (- y + - z) < - x + - x -> ?w:set.w iIn SNoL z & y + w = x + x