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

theorem
  for G, v, V for G1 being addAdjVertexToAll of G,v,V
  for G2 being addAdjVertexFromAll of G,v,V
  for W1 being Walk of G1 holds W1 is Walk of G2
proof
  let G,v,V;
  let G1 be addAdjVertexToAll of G,v,V;
  let G2 be addAdjVertexFromAll of G,v,V;
  let W1 be Walk of G1;
  per cases;
  suppose V c= the_Vertices_of G & not v in the_Vertices_of G;
    then G2 is reverseEdgeDirections of G1, G1.edgesOutOf({v}) by Th35;
    hence thesis by Th14;
  end;
  suppose not (V c= the_Vertices_of G & not v in the_Vertices_of G);
    then G1 == G & G2 == G by Def2, Def3;
    hence thesis by GLIB_000:85, GLIB_001:179;
  end;
end;
