const In_rec_G_ii : (set (set set set) set set) set (set set) prop const In : set set prop term iIn = In infix iIn 2000 2000 axiom In_rec_G_ii_inv: !P:set (set set set) set set.!x:set.!f:set set.In_rec_G_ii P x f -> ?g:set set set.(!y:set.y iIn x -> In_rec_G_ii P y (g y)) & f = P x g axiom In_ind: !p:set prop.(!x:set.(!y:set.y iIn x -> p y) -> p x) -> !x:set.p x lemma !P:set (set set set) set set.!x:set.!f:set set.!f2:set set.(!y:set.!g:set set set.!h:set set set.(!z:set.z iIn y -> g z = h z) -> P y g = P y h) -> (!y:set.y iIn x -> !f3:set set.!f4:set set.In_rec_G_ii P y f3 -> In_rec_G_ii P y f4 -> f3 = f4) -> In_rec_G_ii P x f -> In_rec_G_ii P x f2 -> (?g:set set set.(!y:set.y iIn x -> In_rec_G_ii P y (g y)) & f = P x g) -> f = f2 claim !P:set (set set set) set set.(!x:set.!g:set set set.!h:set set set.(!y:set.y iIn x -> g y = h y) -> P x g = P x h) -> !x:set.!f:set set.!f2:set set.In_rec_G_ii P x f -> In_rec_G_ii P x f2 -> f = f2