reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;

theorem Th89:
  for G2, V for G1 being addVertices of G2, V
  for x,y,e being object holds e DJoins x,y,G1 iff e DJoins x,y,G2
proof
  let G2, V;
  let G1 be addVertices of G2, V;
  let x,y,e be object;
  hereby
    assume e DJoins x,y,G1;
    then e in the_Edges_of G1 &
      (the_Source_of G1).e = x & (the_Target_of G1).e = y by GLIB_000:def 14;
    then e in the_Edges_of G2 &
      (the_Source_of G2).e = x & (the_Target_of G2).e = y by Def10;
    hence e DJoins x,y,G2 by GLIB_000:def 14;
  end;
  assume e DJoins x,y,G2;
  then e in the_Edges_of G2 &
    (the_Source_of G2).e = x & (the_Target_of G2).e = y by GLIB_000:def 14;
  then e in the_Edges_of G1 &
    (the_Source_of G1).e = x & (the_Target_of G1).e = y by Def10;
  hence e DJoins x,y,G1 by GLIB_000:def 14;
end;
