
theorem
  for G2 being _Graph, v1,e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2 holds G1.loops() = G2.loops()
proof
  let G2 be _Graph, v1,e,v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  A1: G2.loops() c= G1.loops() by Th49;
  per cases;
  suppose A2: v1 in the_Vertices_of G2 & not v2 in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    now
      let e0 be object;
      assume e0 in G1.loops();
      then consider w being object such that
        A3: e0 Joins w,w,G1 by Def2;
      e0 in the_Edges_of G2
      proof
        assume A4: not e0 in the_Edges_of G2;
        A5: e0 in the_Edges_of G1 by A3, GLIB_000:def 13;
        the_Edges_of G1 = the_Edges_of G2 \/ {e} by A2, GLIB_006:def 12;
        then e0 in {e} by A4, A5, XBOOLE_0:def 3;
        then e = e0 by TARSKI:def 1;
        then e0 Joins v1,v2,G1 by A2, GLIB_006:131;
        then v1 = w & v2 = w by A3, GLIB_000:15;
        hence contradiction by A2;
      end;
      then e0 Joins w,w,G2 by A3, GLIB_006:72;
      hence e0 in G2.loops() by Def2;
    end;
    then G1.loops() c= G2.loops() by TARSKI:def 3;
    hence thesis by A1, XBOOLE_0:def 10;
  end;
  suppose A6: not v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    now
      let e0 be object;
      assume e0 in G1.loops();
      then consider w being object such that
        A7: e0 Joins w,w,G1 by Def2;
      e0 in the_Edges_of G2
      proof
        assume A8: not e0 in the_Edges_of G2;
        A9: e0 in the_Edges_of G1 by A7, GLIB_000:def 13;
        the_Edges_of G1 = the_Edges_of G2 \/ {e} by A6, GLIB_006:def 12;
        then e0 in {e} by A8, A9, XBOOLE_0:def 3;
        then e = e0 by TARSKI:def 1;
        then e0 Joins v1,v2,G1 by A6, GLIB_006:132;
        then v1 = w & v2 = w by A7, GLIB_000:15;
        hence contradiction by A6;
      end;
      then e0 Joins w,w,G2 by A7, GLIB_006:72;
      hence e0 in G2.loops() by Def2;
    end;
    then G1.loops() c= G2.loops() by TARSKI:def 3;
    hence thesis by A1, XBOOLE_0:def 10;
  end;
  suppose not (v1 in the_Vertices_of G2 & not v2 in the_Vertices_of G2 &
        not e in the_Edges_of G2) & not (not v1 in the_Vertices_of G2 &
        v2 in the_Vertices_of G2 & not e in the_Edges_of G2);
    then G1 == G2 by GLIB_006:def 12;
    hence thesis by Th50;
  end;
end;
