const In_rec_G_iii : (set (set set 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_iii_inv: !P:set (set set set set) set set set.!x:set.!g:set set set.In_rec_G_iii P x g -> ?Q:set set set set.(!y:set.y iIn x -> In_rec_G_iii P y (Q y)) & g = P x Q lemma !P:set (set set set set) set set set.!x:set.!g:set set set.!h:set set set.(!y:set.!Q:set set set set.!R:set set set set.(!z:set.z iIn y -> Q z = R z) -> P y Q = P y R) -> (!y:set.y iIn x -> !g2:set set set.!h2:set set set.In_rec_G_iii P y g2 -> In_rec_G_iii P y h2 -> g2 = h2) -> In_rec_G_iii P x h -> (?Q:set set set set.(!y:set.y iIn x -> In_rec_G_iii P y (Q y)) & g = P x Q) -> (?Q:set set set set.(!y:set.y iIn x -> In_rec_G_iii P y (Q y)) & h = P x Q) -> g = h var P:set (set set set set) set set set var x:set var g:set set set var h:set set set hyp !y:set.!Q:set set set set.!R:set set set set.(!z:set.z iIn y -> Q z = R z) -> P y Q = P y R hyp !y:set.y iIn x -> !g2:set set set.!h2:set set set.In_rec_G_iii P y g2 -> In_rec_G_iii P y h2 -> g2 = h2 hyp In_rec_G_iii P x g hyp In_rec_G_iii P x h claim (?Q:set set set set.(!y:set.y iIn x -> In_rec_G_iii P y (Q y)) & g = P x Q) -> g = h