reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th65:
  VertexDomRel(createGraph(V,E)) = E
proof
  set G0 = createGraph(V,E);
  now
    let v,w be object;
    hereby
      assume [v,w] in VertexDomRel(G0);
      then consider e being object such that
        A1: e DJoins v,w,G0 by Th1;
      e in the_Edges_of G0 by A1, GLIB_000:def 14;
      then A2: e in E;
      then consider v0,w0 being object such that
        A3: e = [v0,w0] by RELAT_1:def 1;
      e DJoins v0,w0,G0 by A2, A3, Th63;
      then v0 = v & w0 = w by A1, GLIB_000:125;
      hence [v,w] in E by A2, A3;
    end;
    assume [v,w] in E;
    then [v,w] DJoins v,w,G0 by Th63;
    hence [v,w] in VertexDomRel(G0) by Th1;
  end;
  hence thesis by RELAT_1:def 2;
end;
