reserve G for _Graph;

theorem
  for H being DLGraphComplement of G holds VertexDomRel(H)
    = [: the_Vertices_of G, the_Vertices_of G :] \ VertexDomRel(G)
proof
  let H be DLGraphComplement of G;
  set N = [: the_Vertices_of G, the_Vertices_of G :];
  now
    let x,y be object;
    hereby
      assume A1: [x,y] in VertexDomRel(H);
      the_Vertices_of G = the_Vertices_of H by GLIB_012:def 6;
      then A2: [x,y] in N by A1;
      consider e2 being object such that
        A3: e2 DJoins x,y,H by A1, Th1;
      x in the_Vertices_of G & y in the_Vertices_of G by A2, ZFMISC_1:87;
      then not ex e1 being object st e1 DJoins x,y,G by A3, GLIB_012:def 6;
      then not [x,y] in VertexDomRel(G) by Th1;
      hence [x,y] in N \ VertexDomRel(G) by A2, XBOOLE_0:def 5;
    end;
    assume [x,y] in N \ VertexDomRel(G);
    then A4: [x,y] in N & not [x,y] in VertexDomRel(G) by XBOOLE_0:def 5;
    then A5: x in the_Vertices_of G & y in the_Vertices_of G by ZFMISC_1:87;
    not ex e1 being object st e1 DJoins x,y,G by A4, Th1;
    then ex e2 being object st e2 DJoins x,y,H by A5, GLIB_012:def 6;
    hence [x,y] in VertexDomRel(H) by Th1;
  end;
  hence thesis by RELAT_1:def 2;
end;
