reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th133:
  for G2 for v1 being Vertex of G2, e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  st not v2 in the_Vertices_of G2 & not e in the_Edges_of G2
  holds v2 is Vertex of G1
proof
  let G2;
  let v1 be Vertex of G2, e,v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume not v2 in the_Vertices_of G2 & not e in the_Edges_of G2;
  then A1: the_Vertices_of G1 = the_Vertices_of G2 \/ {v2} by Def13;
  v2 in {v2} by TARSKI:def 1;
  hence thesis by A1, XBOOLE_0:def 3;
end;
