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

theorem Th49:
  for G2, v, V for G1 being addAdjVertexAll of G2,v,V, v1,e,v2 being object
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 &
    v1 <> v & v2 <> v & e Joins v1,v2,G1
  holds e Joins v1,v2,G2
proof
  let G2, v, V;
  let G1 being addAdjVertexAll of G2,v,V;
  let v1,e,v2 being object;
  assume that
    A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2 and
    A2: v1 <> v & v2 <> v and
    A3: e Joins v1,v2,G1;
  per cases by A3, GLIB_000:16;
  suppose e DJoins v1,v2,G1;
    then e DJoins v1,v2,G2 by A1, A2, Def4;
    hence thesis by GLIB_000:16;
  end;
  suppose e DJoins v2,v1,G1;
    then e DJoins v2,v1,G2 by A1, A2, Def4;
    hence thesis by GLIB_000:16;
  end;
end;
