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

theorem Th134:
  for G2 for v1, e being object, v2 being Vertex of G2
  for G1 being addAdjVertex of G2,v1,e,v2
  st not v1 in the_Vertices_of G2 & not e in the_Edges_of G2
  holds v1 is Vertex of G1
proof
  let G2;
  let v1,e be object, v2 be Vertex of G2;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume not v1 in the_Vertices_of G2 & not e in the_Edges_of G2;
  then A1: the_Vertices_of G1 = the_Vertices_of G2 \/ {v1} by Def14;
  v1 in {v1} by TARSKI:def 1;
  hence thesis by A1, XBOOLE_0:def 3;
end;
