axiom exandE_iiii: !P:(set set set set) prop.!Q:(set set set set) prop.(?R:set set set set.P R & Q R) -> !R:prop.(!P2:set set set set.P P2 -> Q P2 -> R) -> R const In : set set prop term iIn = In infix iIn 2000 2000 const In_rec_G_iii : (set (set set set set) set set set) set (set set set) prop 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 h hyp ?Q:set set set set.(!y:set.y iIn x -> In_rec_G_iii P y (Q y)) & g = P x Q claim (?Q:set set set set.(!y:set.y iIn x -> In_rec_G_iii P y (Q y)) & h = P x Q) -> g = h