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

theorem
  for G2 being _finite _Graph, v being object, V being 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.order() = G2.order() + 1
proof
  let G2 be _finite _Graph, v be object, V be set;
  let G1 be addAdjVertexAll of G2, v, V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  then A2: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by Def4;
  thus G1.order() = card the_Vertices_of G1 by GLIB_000:def 24
    .= card the_Vertices_of G2 + 1 by A1, A2, CARD_2:41
    .= G2.order() + 1 by GLIB_000:def 24;
end;
