
theorem Th21:
  for G1, G2 being _Graph holds G1 is addLoops of G2, {} iff G1 == G2
proof
  let G1, G2 be _Graph;
  hereby
    assume A1: G1 is addLoops of G2, {};
    {} c= the_Vertices_of G2 by XBOOLE_1:2;
    then consider E being set, f being one-to-one Function such that
      A2: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E &
        dom f = E & rng f = {} & the_Source_of G1 = the_Source_of G2 +* f &
        the_Target_of G1 = the_Target_of G2 +* f by A1, Def5;
    A3: f = {} by A2;
    then A4: E = {} by A2;
    A5: the_Source_of G1 = the_Source_of G2 by A2, A3, FUNCT_4:21;
    the_Target_of G1 = the_Target_of G2 by A2, A3, FUNCT_4:21;
    hence G1 == G2 by A1, A2, A4, A5, Th15, GLIB_000:def 34;
  end;
  assume A6: G1 == G2;
  then A7: G1 is Supergraph of G2 by GLIB_006:59;
  now
    thus {} c= the_Vertices_of G2 by XBOOLE_1:2;
    thus the_Vertices_of G1 = the_Vertices_of G2 by A6, GLIB_000:def 34;
    set E = the empty set, f = the empty one-to-one Function;
    take E,f;
    thus E misses the_Edges_of G2 by XBOOLE_1:65;
    thus the_Edges_of G1 = the_Edges_of G2 \/ E by A6, GLIB_000:def 34;
    thus dom f = E & rng f = {};
    thus the_Source_of G1 = the_Source_of G2 by A6, GLIB_000:def 34
      .= the_Source_of G2 +* f by FUNCT_4:21;
    thus the_Target_of G1 = the_Target_of G2 by A6, GLIB_000:def 34
      .= the_Target_of G2 +* f by FUNCT_4:21;
  end;
  hence G1 is addLoops of G2, {} by A7, Def5;
end;
