
theorem
  for G2 being _Graph, V being set, G1 being addVertices of G2, V
  holds G1.loops() = G2.loops()
proof
  let G2 be _Graph, V be set, G1 be addVertices of G2, V;
  now
    let e be object;
    assume e in G1.loops();
    then consider v being object such that
      A1: e Joins v,v,G1 by Def2;
    v is set by TARSKI:1;
    then e Joins v,v,G2 by A1, Th41;
    hence e in G2.loops() by Def2;
  end;
  then A2: G1.loops() c= G2.loops() by TARSKI:def 3;
  G2.loops() c= G1.loops() by Th49;
  hence thesis by A2, XBOOLE_0:def 10;
end;
