reserve G for _Graph;

theorem Th57:
  for v, w being Vertex of G, e being object, H being addEdge of G,v,e,w
  st not e in the_Edges_of G
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G) \/ {[v,w], [w,v]}
proof
  let v,w be Vertex of G, e be object, H be addEdge of G,v,e,w;
  assume A1: not e in the_Edges_of G;
  set R = VertexDomRel(G), U = R \/ {[v,w]};
  A2: U~ = R~ \/ ({[v,w]} qua Relation)~ by RELAT_1:23;
  thus VertexAdjSymRel(H) = U \/ (VertexDomRel(H))~ by A1, Th27
    .= U \/ U~ by A1, Th27
    .= U \/ (R~ \/ {[w,v]}) by A2, GLIBPRE0:12
    .= (U \/ R~) \/ {[w,v]} by XBOOLE_1:4
    .= ((R \/ R~) \/ {[v,w]}) \/ {[w,v]} by XBOOLE_1:4
    .= (R \/ R~) \/ ({[v,w]} \/ {[w,v]}) by XBOOLE_1:4
    .= VertexAdjSymRel(G) \/ {[v,w], [w,v]} by ENUMSET1:1;
end;
