reserve G for _Graph;

theorem Th30:
  for V being Subset of the_Vertices_of G, H being addLoops of G, V
  holds VertexDomRel(H) = VertexDomRel(G) \/ id V
proof
  let V be Subset of the_Vertices_of G, H be addLoops of G,V;
  consider E being set, f being one-to-one Function such that
    A1: E misses the_Edges_of G & the_Edges_of H = the_Edges_of G \/ E and
    A2: dom f = E & rng f = V and
    A3: the_Source_of H = the_Source_of G +* f and
    A4: the_Target_of H = the_Target_of G +* f by GLIB_012:def 5;
  now
    let v,w be object;
    hereby
      assume [v,w] in VertexDomRel(H);
      then consider e being object such that
        A5: e DJoins v,w,H by Th1;
      per cases by A5, GLIB_006:71;
      suppose e DJoins v,w,G;
        then [v,w] in VertexDomRel(G) by Th1;
        hence [v,w] in VertexDomRel(G) \/ id V by XBOOLE_0:def 3;
      end;
      suppose A6: not e in the_Edges_of G;
        e in the_Edges_of H by A5, GLIB_000:def 14;
        then A7: e in E by A1, A6, XBOOLE_0:def 3;
        A8: v = (the_Source_of H).e by A5, GLIB_000:def 14
          .= f.e by A2, A3, A7, FUNCT_4:13;
        w = (the_Target_of H).e by A5, GLIB_000:def 14
          .= f.e by A2, A4, A7, FUNCT_4:13;
        then A9: v = w by A8;
        v in V by A2, A7, A8, FUNCT_1:3;
        then [v,w] in id V by A9, RELAT_1:def 10;
        hence [v,w] in VertexDomRel(G) \/ id V by XBOOLE_0:def 3;
      end;
    end;
    assume [v,w] in VertexDomRel(G) \/ id V;
    then per cases by XBOOLE_0:def 3;
    suppose A10: [v,w] in VertexDomRel(G);
      G is Subgraph of H by GLIB_006:57;
      hence [v,w] in VertexDomRel(H) by A10, Th15, TARSKI:def 3;
    end;
    suppose [v,w] in id V;
      then A11: v = w & v in V by RELAT_1:def 10;
      then reconsider v0 = v as Vertex of H by GLIB_012:def 5;
      v0,v0 are_adjacent by A11, GLIB_012:18;
      then consider e being object such that
        A12: e Joins v,v,H by CHORD:def 3;
      e DJoins v,v,H by A12, GLIB_000:16;
      hence [v,w] in VertexDomRel(H) by A11, Th1;
    end;
  end;
  hence thesis by RELAT_1:def 2;
end;
