
theorem
  for V being non empty set, E being Relation of V
  holds createGraph(V,E) is Dcomplete iff [: V, V :] \ id V c= E
proof
  let V be non empty set, E be Relation of V;
  set G = createGraph(V,E);
  hereby
    assume A1: G is Dcomplete;
    now
      let v,w be Element of V;
      assume [v,w] in [: V, V :] \ id V;
      then not [v,w] in id V by XBOOLE_0:def 5;
      then A2: v <> w by RELAT_1:def 10;
      consider e being object such that
        A3: e DJoins v,w,G by A1, A2;
      e = [v,w] by A3, GLUNIR00:64;
      hence [v,w] in E by A3, GLUNIR00:63;
    end;
    hence [: V, V :] \ id V c= E by RELSET_1:def 1;
  end;
  assume A4: [: V, V :] \ id V c= E;
  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 A4, GLUNIR00:63;
end;
