reserve G for _Graph;

theorem
  for v being Vertex of G, e,w being object, H being addAdjVertex of G,v,e,w
  st not e in the_Edges_of G & not w in the_Vertices_of G
  holds VertexDomRel(H) = VertexDomRel(G) \/ {[v,w]}
proof
  let v be Vertex of G, e,w be object, H be addAdjVertex of G,v,e,w;
  assume A1: not e in the_Edges_of G & not w in the_Vertices_of G;
  then consider G9 being addVertex of G,w such that
    A2: H is addEdge of G9,v,e,w by GLIB_006:125;
  A3: not e in the_Edges_of G9 by A1, GLIB_006:def 10;
  v is Vertex of G9 & w is Vertex of G9 by GLIB_006:68, GLIB_006:94;
  hence VertexDomRel(H) = VertexDomRel(G9) \/ {[v,w]} by A2, A3, Th27
    .= VertexDomRel(G) \/ {[v,w]} by Th25;
end;
