reserve G for _Graph;

theorem Th17:
  for H being removeLoops of G
  holds VertexDomRel(H) = VertexDomRel(G) \ id the_Vertices_of G
proof
  let H be removeLoops of G;
  now
    let v,w be object;
    hereby
      assume A1: [v,w] in VertexDomRel(H);
      then consider e being object such that
        A2: e DJoins v,w,H by Th1;
      v <> w by A2, GLIB_000:136;
      then A3: not [v,w] in id the_Vertices_of G by RELAT_1:def 10;
      [v,w] in VertexDomRel(G) by A1, Th15, TARSKI:def 3;
      hence [v,w] in VertexDomRel(G) \ id the_Vertices_of G
        by A3, XBOOLE_0:def 5;
    end;
    assume [v,w] in VertexDomRel(G) \ id the_Vertices_of G;
    then A4: [v,w] in VertexDomRel(G) & not [v,w] in id the_Vertices_of G
      by XBOOLE_0:def 5;
    then consider e being object such that
      A5: e DJoins v,w,G by Th1;
    A6: e Joins v,w,G by A5, GLIB_000:16;
    then v in the_Vertices_of G by GLIB_000:13;
    then v <> w by A4, RELAT_1:def 10;
    then A7: not e in G.loops() by A6, GLIB_009:46;
    e in the_Edges_of G by A5, GLIB_000:def 14;
    then e in the_Edges_of G \ G.loops() by A7, XBOOLE_0:def 5;
    then A8: e in the_Edges_of H by GLIB_000:53;
    v is set & w is set & e is set by TARSKI:1;
    then e DJoins v,w,H by A5, A8, GLIB_000:73;
    hence [v,w] in VertexDomRel(H) by Th1;
  end;
  hence thesis by RELAT_1:def 2;
end;
