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

theorem
  for G, G2, v, V for G1 being addAdjVertexAll of G,v,V
  st G1 == G2 holds G2 is addAdjVertexAll of G,v,V
proof
  let G, G2, v, V;
  let G1 be addAdjVertexAll 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 consider E being set such that
      A3: card V = card E & E misses the_Edges_of G and
      A4: the_Edges_of G1 = the_Edges_of G \/ E and
      A5: for v1 being object st v1 in V ex e1 being object st e1 in E &
        e1 Joins v1,v,G1 &
        for e2 being object st e2 Joins v1,v,G1 holds e1 = e2 by Def4;
    :: show all properties of Definition
    A6: now
      the_Vertices_of G1 = the_Vertices_of G \/ {v} by A2, Def4;
      hence the_Vertices_of G2 = the_Vertices_of G \/ {v}
        by A1, GLIB_000:def 34;
      hereby
        let e be object;
        not e Joins v,v,G1 by A2, Def4;
        hence not e Joins v,v,G2 by A1, GLIB_000:88;
        let v1 be object;
        hereby
          assume not v1 in V;
          then not e Joins v1,v,G1 by A2, Def4;
          hence not e Joins v1,v,G2 by A1, GLIB_000:88;
        end;
        let v2 be object;
        assume that
          A7: v1 <> v & v2 <> v and
          A8: e DJoins v1,v2,G2;
        e DJoins v1,v2,G1 by A8, A1, GLIB_000:88;
        hence e DJoins v1,v2,G by A2, A7, Def4;
      end;
      take E;
      thus card V = card E & E misses the_Edges_of G by A3;
      thus the_Edges_of G2 = the_Edges_of G \/ E by A4, A1, GLIB_000:def 34;
      let v1 be object;
      assume v1 in V;
      then consider e1 being object such that
        A9: e1 in E & e1 Joins v1,v,G1 and
        A10: for e2 being object st e2 Joins v1,v,G1 holds e1 = e2 by A5;
      take e1;
      thus e1 in E & e1 Joins v1,v,G2 by A1, A9, GLIB_000:88;
      let e2 be object;
      assume e2 Joins v1,v,G2;
      then e2 Joins v1,v,G1 by A1, GLIB_000:88;
      hence e1 = e2 by A10;
    end;
    G2 is Supergraph of G1 by A1, GLIB_006:59;
    then G2 is Supergraph of G by GLIB_006:62;
    hence thesis by A2, A6, Def4;
  end;
  suppose A11: not (V c= the_Vertices_of G & not v in the_Vertices_of G);
    then G == G1 by Def4;
    then A12: G == G2 by A1, GLIB_000:85;
    then G2 is Supergraph of G by GLIB_006:59;
    hence thesis by A11, A12, Def4;
  end;
end;
