
theorem
  for G2 being _Graph, v being object, G1 being addVertex of G2, v
  holds G1 is addAdjVertexAll of G2,v,{}
proof
  let G2 be _Graph, v be object, G1 be addVertex of G2, v;
  per cases;
  suppose A1: not v in the_Vertices_of G2;
    A2: {} c= the_Vertices_of G2 by XBOOLE_1:2;
    now
      thus the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by GLIB_006:def 10;
      hereby
        let e be object;
        not e Joins v,v,G2 by A1, GLIB_000:13;
        hence not e Joins v,v,G1 by GLIB_006:87;
        let v1 be object;
        hereby
          assume not v1 in {};
          not e Joins v1,v,G2 by A1, GLIB_000:13;
          hence not e Joins v1,v,G1 by GLIB_006:87;
        end;
        let v2 be object;
        assume v1 <> v & v2 <> v & e DJoins v1,v2,G1;
        hence e DJoins v1,v2,G2 by GLIB_006:85;
      end;
      take E = {};
      thus card {} = card E;
      thus E misses the_Edges_of G2 by XBOOLE_1:65;
      thus the_Edges_of G1 = the_Edges_of G2 \/ E by GLIB_006:def 10;
      let v1 be object;
      assume v1 in {};
      hence 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;
    end;
    hence thesis by A1, A2, GLIB_007:def 4;
  end;
  suppose A3: v in the_Vertices_of G2;
    then G1 == G2 by ZFMISC_1:31, GLIB_006:78;
    hence thesis by A3, GLIB_007:def 4;
  end;
end;
