
theorem Th80:
  for G being _finite simple _Graph
  ex p being non empty _finite simple Graph-yielding FinSequence
  st p.1 is _trivial edgeless & p.len p = G & len p = G.order() &
    for n being Element of dom p st n <= len p - 1
    ex v being object, V being finite set
    st v in the_Vertices_of G \ the_Vertices_of p.n &
      V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V
proof
  let G be _finite simple _Graph;
  set v0 = the Vertex of G;
  set H = the inducedSubgraph of G,{v0};
  consider p being non empty _finite simple Graph-yielding FinSequence
  such that
    A1: p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 and
    A2: for n being Element of dom p st n <= len p - 1
      ex v being object, V being finite set
      st v in the_Vertices_of G \ the_Vertices_of p.n &
        V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V
    by Th79;
  take p;
  thus p.1 is _trivial edgeless by A1, Th52, GLIB_000:89;
  thus p.len p = G by A1;
  H.order() = 1 by GLIB_000:26;
  hence len p = G.order() by A1;
  thus thesis by A2;
end;
