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

theorem Th32:
  for G, v, V for G1 being addAdjVertexToAll of G,v,V
  for G2 being addAdjVertexFromAll of G,v,V
  holds the_Vertices_of G1 = the_Vertices_of G2
    & the_Edges_of G1 = the_Edges_of G2
proof
  let G,v,V;
  let G1 be addAdjVertexToAll of G,v,V;
  let G2 be addAdjVertexFromAll of G,v,V;
  per cases;
  suppose A1: V c= the_Vertices_of G & not v in the_Vertices_of G;
    then the_Vertices_of G1 = the_Vertices_of G \/ {v}
      & the_Edges_of G1 = the_Edges_of G \/ (V --> the_Edges_of G) by Def2;
    hence thesis by A1,Def3;
  end;
  suppose not (V c= the_Vertices_of G & not v in the_Vertices_of G);
    then G == G1 & G == G2 by Def2, Def3;
    then G1 == G2 by GLIB_000:85;
    hence thesis by GLIB_000:def 34;
  end;
end;
