const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set const If_i : prop set set set const SNo_rec_i : (set (set set) set) set set lemma !P:set set (set set set) set.!x:set.!g:set set set.(!y:set.SNo y -> !z:set.SNo z -> !h:set set set.!g2:set set set.(!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.SNo u -> h w u = g2 w u) -> (!w:set.w iIn SNoS_ (SNoLev z) -> h y w = g2 y w) -> P y z h = P y z g2) -> SNo x -> (!y:set.SNo y -> !f:set set.!f2:set set.(!z:set.z iIn SNoS_ (SNoLev y) -> f z = f2 z) -> P x y (\z:set.\w:set.If_i (z = x) (f w) (g z w)) = P x y \z:set.\w:set.If_i (z = x) (f2 w) (g z w)) -> !y:set.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 = P x y \z:set.\w:set.If_i (z = x) (SNo_rec_i (\u:set.\f:set set.P x u \v:set.\x2:set.If_i (v = x) (f x2) (g v x2)) w) (g z w) lemma !P:set set (set set set) set.!x:set.!g:set set set.!y:set.!f:set set.!f2:set set.(!z:set.SNo z -> !w:set.SNo w -> !h:set set set.!g2:set set set.(!u:set.u iIn SNoS_ (SNoLev z) -> !v:set.SNo v -> h u v = g2 u v) -> (!u:set.u iIn SNoS_ (SNoLev w) -> h z u = g2 z u) -> P z w h = P z w g2) -> SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev y) -> f z = f2 z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> g z = g z) -> P x y (\z:set.\w:set.If_i (z = x) (f w) (g z w)) = P x y \z:set.\w:set.If_i (z = x) (f2 w) (g z w) claim !P:set set (set set set) set.(!x:set.SNo x -> !y:set.SNo y -> !g:set set set.!h:set set set.(!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.SNo w -> g z w = h z w) -> (!z:set.z iIn SNoS_ (SNoLev y) -> g x z = h x z) -> P x y g = P x y h) -> !x:set.SNo x -> !g:set set set.!y:set.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 = P x y \z:set.\w:set.If_i (z = x) (SNo_rec_i (\u:set.\f:set set.P x u \v:set.\x2:set.If_i (v = x) (f x2) (g v x2)) w) (g z w)