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

theorem
  for G2, v, V for G1 being addAdjVertexAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds G1.edgesBetween(V,{v}) misses the_Edges_of G2 &
    the_Edges_of G1 = the_Edges_of G2 \/ G1.edgesBetween(V,{v})
proof
  let G2, v, V;
  let G1 be addAdjVertexAll of G2, v, V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  G1.edgesBetween(V,{v}) /\ the_Edges_of G2 = {}
  proof
    assume A2: G1.edgesBetween(V,{v}) /\ the_Edges_of G2 <> {};
    set e = the Element of G1.edgesBetween(V,{v}) /\ the_Edges_of G2;
    A3: e in G1.edgesBetween(V,{v}) /\ the_Edges_of G2 by A2;
    e in G1.edgesBetween(V,{v}) by XBOOLE_0:def 4,A2;
    then e SJoins V,{v},G1 by GLIB_000:def 30;
    then consider w being object such that
      A4: w in V & e Joins w,v,G1 by GLIB_000:102;
    e in the_Edges_of G2 by A3, XBOOLE_0:def 4;
    then e Joins w,v,G2 by A4, GLIB_006:72;
    hence contradiction by A1, GLIB_000:13;
  end;
  hence G1.edgesBetween(V,{v}) misses the_Edges_of G2 by XBOOLE_0:def 7;
  A5: the_Edges_of G2 c= the_Edges_of G1 by GLIB_006:def 9;
  A6: the_Edges_of G2 \/ G1.edgesBetween(V,{v}) c= the_Edges_of G1
    by A5, XBOOLE_1:8;
  for e being object holds e in the_Edges_of G1 implies
    e in the_Edges_of G2 \/ G1.edgesBetween(V,{v})
  proof
    let e be object;
    set v1 = (the_Source_of G1).e, v2 = (the_Target_of G1).e;
    assume e in the_Edges_of G1;
    then A7: e Joins v1,v2,G1 by GLIB_000:def 13;
    e in the_Edges_of G2 or e in G1.edgesBetween(V,{v})
    proof
      assume A8: not e in the_Edges_of G2;
      consider E being set such that
        A9: card V = card E & E misses the_Edges_of G2 &
        the_Edges_of G1 = the_Edges_of G2 \/ E and
        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 A1, Def4;
      per cases by A1, A9, A7, A8, Th51;
      suppose v1 = v & v2 in V;
        then v1 in {v} & v2 in V by TARSKI:def 1;
        then e SJoins V,{v},G1 by A7, GLIB_000:17;
        hence thesis by GLIB_000:def 30;
      end;
      suppose v2 = v & v1 in V;
        then v2 in {v} & v1 in V by TARSKI:def 1;
        then e SJoins V,{v},G1 by A7, GLIB_000:17;
        hence thesis by GLIB_000:def 30;
      end;
    end;
    hence thesis by XBOOLE_0:def 3;
  end;
  then the_Edges_of G1 c= the_Edges_of G2 \/ G1.edgesBetween(V,{v})
    by TARSKI:def 3;
  hence thesis by A6, XBOOLE_0:def 10;
end;
