axiom exandE_iii: !P:(set set set) prop.!Q:(set set set) prop.(?g:set set set.P g & Q g) -> !R:prop.(!g:set set set.P g -> Q g -> R) -> R const In : set set prop term iIn = In infix iIn 2000 2000 const In_rec_G_ii : (set (set set set) set set) set (set set) prop var P:set (set set set) set set var x:set var f:set set var f2:set set hyp !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 hyp !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 hyp In_rec_G_ii P x f2 hyp ?g:set set set.(!y:set.y iIn x -> In_rec_G_ii P y (g y)) & f = P x g claim (?g:set set set.(!y:set.y iIn x -> In_rec_G_ii P y (g y)) & f2 = P x g) -> f = f2