reserve G for _Graph;

theorem Th22:
  for V being set, H being removeVertices of G, V st V c< the_Vertices_of G
  holds VertexDomRel(H) = VertexDomRel(G)
    \ ([: V, the_Vertices_of G :] \/ [: the_Vertices_of G, V :])
proof
  let V be set, H be removeVertices of G, V;
  assume V c< the_Vertices_of G;
  then A1: the_Vertices_of H = the_Vertices_of G \ V &
    the_Edges_of H = G.edgesBetween(the_Vertices_of G \ V) by GLIB_000:49;
  now
    let v,w be object;
    hereby
      assume A2: [v,w] in VertexDomRel(H);
      then A3: [v,w] in VertexDomRel(G) by Th15, TARSKI:def 3;
      consider e being object such that
        A4: e DJoins v,w,H by A2, Th1;
      e Joins v,w,H by A4, GLIB_000:16;
      then v in the_Vertices_of H & w in the_Vertices_of H by GLIB_000:13;
      then not v in V & not w in V by A1, XBOOLE_0:def 5;
      then not [v,w] in [: V, the_Vertices_of G :] &
        not [v,w] in [: the_Vertices_of G, V :] by ZFMISC_1:87;
      then not [v,w] in [: V, the_Vertices_of G :]\/[: the_Vertices_of G, V :]
        by XBOOLE_0:def 3;
      hence [v,w] in VertexDomRel(G) \ ([: V, the_Vertices_of G :] \/
        [: the_Vertices_of G, V :]) by A3, XBOOLE_0:def 5;
    end;
    assume [v,w] in VertexDomRel(G) \ ([: V, the_Vertices_of G :] \/
      [: the_Vertices_of G, V :]);
    then A5: [v,w] in VertexDomRel(G) &
      not [v,w] in [: V, the_Vertices_of G :]\/[: the_Vertices_of G, V :]
      by XBOOLE_0:def 5;
    then A6: not [v,w] in [: V, the_Vertices_of G :] &
      not [v,w] in [: the_Vertices_of G, V :] by XBOOLE_0:def 3;
    consider e being object such that
      A7: e DJoins v,w,G by A5, Th1;
    A8: e Joins v,w,G by A7, GLIB_000:16;
    then A9: v in the_Vertices_of G & w in the_Vertices_of G by GLIB_000:13;
    then not v in V & not w in V by A6, ZFMISC_1:87;
    then v in the_Vertices_of G \ V & w in the_Vertices_of G \ V
      by A9, XBOOLE_0:def 5;
    then A10: e in the_Edges_of H by A1, A8, GLIB_000:32;
    e is set & v is set & w is set by TARSKI:1;
    then e DJoins v,w,H by A7, A10, GLIB_000:73;
    hence [v,w] in VertexDomRel(H) by Th1;
  end;
  hence thesis by RELAT_1:def 2;
end;
