reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem
  for G2 being _finite _Graph, v being object, V being finite set
  for G1 being addAdjVertexAll of G2, v, V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds G1.size() = G2.size() + card V
proof
  let G2 be _finite _Graph, v be object, V be finite set;
  let G1 be addAdjVertexAll of G2, v, V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  consider E being set such that
    A2: card V = card E and
    A3: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E and
    for w being object st w in V ex e1 being object st e1 in E &
      e1 Joins w,v,G1 &
      for e2 being object st e2 Joins w,v,G1 holds e1 = e2
    by A1, Def4;
  V,E are_equipotent by A2, CARD_1:5;
  then reconsider E as finite set by CARD_1:38;
  thus G1.size() = card the_Edges_of G1 by GLIB_000:def 25
    .= card the_Edges_of G2 + card E by A3, CARD_2:40
    .= G2.size() + card V by A2, GLIB_000:def 25;
end;
