
theorem
  for G2 being _Graph, v being object
  for V being finite set, G1 being addAdjVertexAll of G2, v, V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  ex p being non empty Graph-yielding FinSequence
  st p.1 = G2 & p.len p = G1 & len p = card V + 2 & p.2 is addVertex of G2,v &
    for n being Element of dom p st 2 <= n & n <= len p - 1 holds
    ex w being Vertex of G2, e being object
    st e in the_Edges_of G1 \ the_Edges_of p.n &
      (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v)
proof
  let G2 be _Graph, v be object, V be finite set;
  let G1 be addAdjVertexAll of G2, v, V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  V = V \/ {} & {} misses V by XBOOLE_1:65;
  then consider p being non empty Graph-yielding FinSequence such that
    A2: p.1 = G2 & p.len p = G1 & len p = card V + 2 and
    A3: p.2 is addAdjVertexAll of G2, v, {} and
    A4: for n being Element of dom p st 2 <= n & n <= len p - 1 holds
      ex w being Vertex of G2, e being object
      st e in the_Edges_of G1 \ the_Edges_of p.n &
        (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v)
    by A1, Th76;
  take p;
  thus p.1 = G2 & p.len p = G1 & len p = card V + 2 by A2;
  thus p.2 is addVertex of G2, v by A3, GLIB_007:55;
  thus thesis by A4;
end;
