const SNo_rec_ii : (set (set set set) set set) set set set const If_i : prop set set set const SNo : set prop const SNo_rec_i : (set (set set) set) set set const Empty : set term SNo_rec2 = \P:set set (set set set) set.SNo_rec_ii \x:set.\g:set set set.\y:set.If_i (SNo y) (SNo_rec_i (\z:set.\f:set set.P x z \w:set.\u:set.If_i (w = x) (f u) (g w u)) y) Empty const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set axiom SNo_rec_ii_eq: !P:set (set set set) set set.(!x:set.SNo x -> !g:set set set.!h:set set set.(!y:set.y iIn SNoS_ (SNoLev x) -> g y = h y) -> P x g = P x h) -> !x:set.SNo x -> SNo_rec_ii P x = P x (SNo_rec_ii P) var P:set set (set set set) set var x:set var y:set hyp !z:set.SNo z -> !w:set.SNo w -> !g:set set set.!h:set set set.(!u:set.u iIn SNoS_ (SNoLev z) -> !v:set.SNo v -> g u v = h u v) -> (!u:set.u iIn SNoS_ (SNoLev w) -> g z u = h z u) -> P z w g = P z w h hyp SNo x hyp SNo y hyp !z:set.SNo z -> !g:set set set.!h:set set set.(!w:set.w iIn SNoS_ (SNoLev z) -> g w = h w) -> (\w:set.If_i (SNo w) (SNo_rec_i (\u:set.\f:set set.P z u \v:set.\x2:set.If_i (v = z) (f x2) (g v x2)) w) Empty) = \w:set.If_i (SNo w) (SNo_rec_i (\u:set.\f:set set.P z u \v:set.\x2:set.If_i (v = z) (f x2) (h v x2)) w) Empty claim If_i (SNo y) (SNo_rec_i (\z:set.\f:set set.P x z \w:set.\u:set.If_i (w = x) (f u) (SNo_rec_ii (\v:set.\g:set set set.\x2:set.If_i (SNo x2) (SNo_rec_i (\y2:set.\f2:set set.P v y2 \z2:set.\w2:set.If_i (z2 = v) (f2 w2) (g z2 w2)) x2) Empty) w u)) y) Empty = P x y (SNo_rec2 P) -> SNo_rec_ii (\z:set.\g:set set set.\w:set.If_i (SNo w) (SNo_rec_i (\u:set.\f:set set.P z u \v:set.\x2:set.If_i (v = z) (f x2) (g v x2)) w) Empty) x y = P x y (SNo_rec2 P)