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

theorem Th50:
  for G2, v, V for G1 being addAdjVertexAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds v is Vertex of G1
proof
  let G2, v, V;
  let G1 be addAdjVertexAll of G2,v,V;
  assume V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  then A1: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by Def4;
  v in {v} by TARSKI:def 1;
  hence thesis by A1, XBOOLE_0:def 3;
end;
