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_max_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> z <= y const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const omega : set term finite = \x:set.?y:set.y iIn omega & equip x y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const ordsucc : set set const Repl : set (set set) set lemma !x:set.!y:set.!f:set set.!z:set.(!w:set.w iIn y -> SNo w) -> (!w:set.w iIn y -> ?u:set.u iIn ordsucc (ordsucc x) & f u = w) -> Subq (Repl (ordsucc x) f) y -> z iIn Repl (ordsucc x) (\w:set.f w) -> SNo z -> (!w:set.w iIn Repl (ordsucc x) f -> SNo w -> w <= z) -> f (ordsucc x) iIn y -> SNo (f (ordsucc x)) -> ?w:set.SNo_max_of y w var x:set var y:set var f:set set var z:set hyp !w:set.w iIn y -> SNo w hyp !w:set.w iIn ordsucc (ordsucc x) -> f w iIn y hyp !w:set.w iIn y -> ?u:set.u iIn ordsucc (ordsucc x) & f u = w hyp Subq (Repl (ordsucc x) f) y hyp z iIn Repl (ordsucc x) \w:set.f w hyp SNo z hyp !w:set.w iIn Repl (ordsucc x) f -> SNo w -> w <= z claim f (ordsucc x) iIn y -> ?w:set.SNo_max_of y w