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

theorem
  for G, G2, v, V for G1 being addAdjVertexToAll of G,v,V
  st G1 == G2 holds G2 is addAdjVertexToAll of G,v,V
proof
  let G, G2, v, V;
  let G1 be addAdjVertexToAll of G,v,V;
  assume A1: G1 == G2;
  per cases;
  suppose A2: 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) &
      the_Source_of G1 = the_Source_of G +* ((V --> the_Edges_of G) --> v) &
      the_Target_of G1 = the_Target_of G +* pr1(V,{the_Edges_of G}) by Def2;
    then A3: the_Vertices_of G2 = the_Vertices_of G \/ {v} &
      the_Edges_of G2 = the_Edges_of G \/ (V --> the_Edges_of G) &
      the_Source_of G2 = the_Source_of G +* ((V --> the_Edges_of G) --> v) &
      the_Target_of G2 = the_Target_of G +* pr1(V,{the_Edges_of G})
      by A1, GLIB_000:def 34;
    G2 is Supergraph of G1 by A1, GLIB_006:59;
    then G2 is Supergraph of G by GLIB_006:62;
    hence thesis by A2, A3, Def2;
  end;
  suppose A4: not (V c= the_Vertices_of G & not v in the_Vertices_of G);
    then G == G1 by Def2;
    then A5: G == G2 by A1, GLIB_000:85;
    then G2 is Supergraph of G by GLIB_006:59;
    hence thesis by A4, A5, Def2;
  end;
end;
