
theorem Th50:
  for G1, G2 being _Graph
  st the_Vertices_of G2 c= the_Vertices_of G1 &
    (for e,x,y being object st e DJoins x,y,G2 holds e DJoins x,y,G1)
  holds G2 is Subgraph of G1 & G1 is Supergraph of G2
proof
  let G1, G2 be _Graph;
  assume that A1: the_Vertices_of G2 c= the_Vertices_of G1 and
    A2: for e,x,y being object st e DJoins x,y,G2 holds e DJoins x,y,G1;
  now
    let e be object;
    assume e in the_Edges_of G2;
    then e DJoins (the_Source_of G2).e,(the_Target_of G2).e,G1
      by A2, GLIB_000:def 14;
    hence e in the_Edges_of G1 by GLIB_000:def 14;
  end;
  then A3: the_Edges_of G2 c= the_Edges_of G1 by TARSKI:def 3;
  now
    let e be set;
    assume e in the_Edges_of G2;
    then e DJoins (the_Source_of G2).e,(the_Target_of G2).e,G1
      by A2, GLIB_000:def 14;
    hence (the_Source_of G2).e = (the_Source_of G1).e &
      (the_Target_of G2).e = (the_Target_of G1).e by GLIB_000:def 14;
  end;
  hence G2 is Subgraph of G1 by A1, A3, GLIB_000:def 32;
  hence thesis by GLIB_006:57;
end;
