
theorem Th72:
  for G being _finite Tree-like _Graph
  ex p being non empty _finite Tree-like 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 v1,v2 being Vertex of G, e being object
    st p.(n+1) is addAdjVertex of p.n,v1,e,v2 &
      e in the_Edges_of G \ the_Edges_of p.n &
      ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or
        (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n))
proof
  let G be _finite Tree-like _Graph;
  set H = the _trivial Subgraph of G;
  consider p being non empty _finite Tree-like 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 v1,v2 being Vertex of G, e being object
      st p.(n+1) is addAdjVertex of p.n,v1,e,v2 &
        e in the_Edges_of G \ the_Edges_of p.n &
        ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or
          (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) by Th71;
  take p;
  thus p.1 is _trivial edgeless by A1, Th52, GLIB_000:89;
  thus p.len p = G by A1;
  thus len p = G.order() - 1 + 1 by A1, GLIB_000:26 .= G.order();
  thus thesis by A2;
end;
