
theorem Th17:
  for G2 being _Graph, V being set, G1 being addLoops of G2, V
  for e,v,w being object st v <> w holds e Joins v,w,G1 iff e Joins v,w,G2
proof
  let G2 be _Graph, V be set, G1 be addLoops of G2, V;
  let e,v,w be object;
  assume A1: v <> w;
  hereby
    assume e Joins v,w,G1;
    then per cases by GLIB_000:16;
    suppose e DJoins v,w,G1;
      then e DJoins v,w,G2 by A1, Th16;
      hence e Joins v,w,G2 by GLIB_000:16;
    end;
    suppose e DJoins w,v,G1;
      then e DJoins w,v,G2 by A1, Th16;
      hence e Joins v,w,G2 by GLIB_000:16;
    end;
  end;
  assume A2: e Joins v,w,G2;
  v is set & w is set by TARSKI:1;
  hence e Joins v,w,G1 by A2, GLIB_006:70;
end;
