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

theorem Th55:
  for G2, v for G1 being addAdjVertexAll of G2,v,{}
  holds G1 is addVertex of G2, v
proof
  let G2, v;
  let G1 be addAdjVertexAll of G2,v,{};
  per cases;
  suppose {} c= the_Vertices_of G2 & not v in the_Vertices_of G2;
    then A1: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by Def4;
    A2: the_Edges_of G1 = the_Edges_of G2 by Th46;
    then A3: dom the_Source_of G1 = the_Edges_of G2 &
      dom the_Target_of G1 = the_Edges_of G2 by GLIB_000:4;
    A4: the_Source_of G1 = (the_Source_of G1) | the_Edges_of G2 by A3
      .= the_Source_of G2 by GLIB_006:69;
    the_Target_of G1 = (the_Target_of G1) | the_Edges_of G2 by A3
      .= the_Target_of G2 by GLIB_006:69;
    hence thesis by A1, A2, A4, GLIB_006:def 10;
  end;
  suppose A5: not ({} c= the_Vertices_of G2 & not v in the_Vertices_of G2);
    then A6: {v} c= the_Vertices_of G2 by XBOOLE_1:2, ZFMISC_1:31;
    G1 == G2 by A5, Def4;
    hence thesis by A6, GLIB_006:79;
  end;
end;
