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 term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const In_rec_iii : (set (set set set set) set set set) set set set set const If_iii : prop (set set set) (set set set) set set set const If_ii : prop (set set) (set set) set set const SNoS_ : set set const ordsucc : set set const SNoLev : set set const Descr_ii : ((set set) prop) set set term SNo_rec_ii = \P:set (set set set) set set.\x:set.In_rec_iii (\y:set.\Q:set set set set.If_iii (ordinal y) (\z:set.If_ii (z iIn SNoS_ (ordsucc y)) (P z \w:set.Q (SNoLev w) w) (Descr_ii \f:set set.True)) \z:set.Descr_ii \f:set set.True) (SNoLev x) x term nIn = \x:set.\y:set.~ x iIn y var x:set var P:set set set set var Q:set set set set var y:set var z:set hyp !w:set.w iIn x -> P w = Q w hyp ordinal x hyp SNoLev y iIn ordsucc x hyp SNoLev z iIn SNoLev y claim SNoLev z iIn x -> P (SNoLev z) z = Q (SNoLev z) z