reserve G for _Graph;

theorem
  for v, w being Vertex of G, e being object, H being addEdge of G,v,e,w
  st v,w are_adjacent holds VertexAdjSymRel(H) = VertexAdjSymRel(G)
proof
  let v,w be Vertex of G, e be object, H be addEdge of G,v,e,w;
  assume v,w are_adjacent;
  then consider e0 being object such that
    A1: e0 Joins v,w,G by CHORD:def 3;
  per cases;
  suppose A2: not e in the_Edges_of G;
    per cases by A1, GLIB_000:16;
    suppose A3: e0 DJoins v,w,G;
      thus VertexAdjSymRel(H)
         = VertexDomRel(G) \/ (VertexDomRel(H))~ by A3, Th26
        .= VertexAdjSymRel(G) by A3, Th26;
    end;
    suppose e0 DJoins w,v,G;
      then A4: [w,v] in VertexDomRel(G) by Th1;
      then A5: [v,w] in (VertexDomRel(G))~ by RELAT_1:def 7;
      set R = VertexDomRel(G), U = R \/ {[v,w]};
      A6: U~ = R~ \/ ({[v,w]} qua Relation)~ by RELAT_1:23;
      thus VertexAdjSymRel(H) = U \/ (VertexDomRel(H))~ by A2, Th27
        .= U \/ U~ by A2, Th27
        .= U \/ (R~ \/ {[w,v]}) by A6, GLIBPRE0:12
        .= (U \/ {[w,v]}) \/ R~ by XBOOLE_1:4
        .= ((R \/ {[w,v]}) \/ {[v,w]}) \/ R~ by XBOOLE_1:4
        .= (R \/ {[w,v]}) \/ (R~ \/ {[v,w]}) by XBOOLE_1:4
        .= R \/ (R~ \/ {[v,w]}) by A4, ZFMISC_1:31, XBOOLE_1:12
        .= VertexAdjSymRel(G) by A5, ZFMISC_1:31, XBOOLE_1:12;
    end;
  end;
  suppose e in the_Edges_of G;
    then G == H by GLIB_006:def 11;
    hence thesis by Th49;
  end;
end;
