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 v1,e,v2 being object
  holds (e Joins v1,v2,G1 iff e Joins v1,v2,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 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 Th9;
  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_000:88;
  end;
end;
