
theorem
  for G being _Graph holds G is Dcomplete iff
    [: the_Vertices_of G, the_Vertices_of G :] \ id the_Vertices_of G
      c= VertexDomRel(G)
proof
  let G be _Graph;
  set V = the_Vertices_of G;
  hereby
    assume A1: G is Dcomplete;
    now
      let v,w be Vertex of G;
      assume [v,w] in [: V, V :] \ id V;
      then not [v,w] in id V by XBOOLE_0:def 5;
      then v <> w by RELAT_1:def 10;
      then ex e being object st e DJoins v,w,G by A1;
      hence [v,w] in VertexDomRel(G) by GLUNIR00:1;
    end;
    hence [: V, V :] \ id V c= VertexDomRel(G) by RELSET_1:def 1;
  end;
  assume A2: [: V, V :] \ id V c= VertexDomRel(G);
  let v,w be Vertex of G;
  assume v <> w;
  then not [v,w] in id V by RELAT_1:def 10;
  then [v,w] in [: V, V :] \ id V by XBOOLE_0:def 5;
  hence thesis by A2, GLUNIR00:1;
end;
