
theorem Th16:
  for G2 being _Graph, V being set, G1 being addLoops of G2, V
  for e,v,w being object st v <> w holds e DJoins v,w,G1 iff e DJoins v,w,G2
proof
  let G2 be _Graph, V be set, G1 be addLoops of G2, V;
  per cases;
  suppose V c= the_Vertices_of G2;
    then consider E being set, f being one-to-one Function such that
      A1: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E &
        dom f = E & rng f = V & the_Source_of G1 = the_Source_of G2 +* f &
        the_Target_of G1 = the_Target_of G2 +* f by Def5;
    let e,v,w be object;
    assume A2: v <> w;
    hereby
      assume A3: e DJoins v,w,G1;
      then A4: e in the_Edges_of G2 \/ E by A1, GLIB_000:def 14;
      not e in E
      proof
        assume A5: e in E;
        then A6: (the_Source_of G1).e = f.e by A1, FUNCT_4:13;
        (the_Target_of G1).e = f.e by A1, A5, FUNCT_4:13;
        then f.e = v & f.e = w by A3, A6, GLIB_000:def 14;
        hence contradiction by A2;
      end;
      then e in the_Edges_of G2 by A1, A4, XBOOLE_0:5;
      hence e DJoins v,w,G2 by A3, GLIB_006:71;
    end;
    assume A7: e DJoins v,w,G2;
    v is set & w is set by TARSKI:1;
    hence thesis by A7, GLIB_006:70;
  end;
  suppose not V c= the_Vertices_of G2;
    then G1 == G2 by Def5;
    hence thesis by GLIB_000:88;
  end;
end;
