reserve G for _Graph;

theorem
  for V being Subset of the_Vertices_of G, H being addLoops of G, V
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G) \/ id V
proof
  let V be Subset of the_Vertices_of G, H be addLoops of G, V;
  set R = VertexDomRel(G);
  reconsider I = id V as Relation;
  thus VertexAdjSymRel(H) = (R \/ I) \/ (VertexDomRel(H))~ by Th30
    .= (R \/ I) \/ (R \/ I)~ by Th30
    .= (R \/ I) \/ (R~ \/ I~) by RELAT_1:23
    .= (R \/ (R~ \/ I)) \/ I by XBOOLE_1:4
    .= (R \/ R~ \/ I) \/ I by XBOOLE_1:4
    .= (R \/ R~) \/ (I \/ I) by XBOOLE_1:4
    .= VertexAdjSymRel(G) \/ id V;
end;
