
theorem Th69:
  for G being _finite _Graph
  ex p being non empty _finite Graph-yielding FinSequence
  st p.1 is _trivial edgeless & p.len p = G & len p = G.order() + G.size() &
    for n being Element of dom p st n <= len p - 1 holds
    (ex v1,v2 being Vertex of G, e being object
     st p.(n+1) is addEdge of p.n,v1,e,v2 &
       e in the_Edges_of G \ the_Edges_of p.n &
       v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) or
    (ex v being Vertex of G st p.(n+1) is addVertex of p.n, v &
      not v in the_Vertices_of p.n)
proof
  let G be _finite _Graph;
  set H = the _trivial edgeless Subgraph of G;
  consider p being non empty _finite Graph-yielding FinSequence such that
    A1: p.1 == H & p.len p = G and
    A2: len p = G.order() + G.size() - (H.order() + H.size()) + 1 and
    A3: for n being Element of dom p st n <= len p - 1 holds
      (ex v1,v2 being Vertex of G, e being object
       st p.(n+1) is addEdge of p.n,v1,e,v2 &
         e in the_Edges_of G \ the_Edges_of p.n &
         v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) or
      (ex v being Vertex of G st p.(n+1) is addVertex of p.n, v &
        not v in the_Vertices_of p.n) by Th68;
  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() + G.size() - (H.order() + 0) + 1 by A2, Th49
    .= G.order() + G.size() - 1 + 1 by GLIB_000:26
    .= G.order() + G.size();
  thus thesis by A3;
end;
