
theorem
  for G2 being _Graph, v being object, V being set
  for G1 being addAdjVertexAll of G2,v,V holds G1.loops() = G2.loops()
proof
  let G2 be _Graph, v be object, V be set;
  let G1 be addAdjVertexAll of G2,v,V;
  per cases;
  suppose A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
    A2: G2.loops() c= G1.loops() by Th49;
    now
      let e be object;
      assume e in G1.loops();
      then consider w being object such that
        A3: e Joins w,w,G1 by Def2;
      v <> w by A1, A3, GLIB_007:def 4;
      then e Joins w,w,G2 by A1, A3, GLIB_007:49;
      hence e in G2.loops() by Def2;
    end;
    then G1.loops() c= G2.loops() by TARSKI:def 3;
    hence thesis by A2, XBOOLE_0:def 10;
  end;
  suppose not (V c= the_Vertices_of G2 & not v in the_Vertices_of G2);
    then G1 == G2 by GLIB_007:def 4;
    hence thesis by Th50;
  end;
end;
