
theorem
  for G being _finite _Graph
  ex p being non empty _finite Graph-yielding FinSequence
  st p.1 is edgeless & p.len p = G & len p = G.size() + 1 &
    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
proof
  let G be _finite _Graph;
  set H = the edgeless spanning Subgraph of G;
  consider p being non empty _finite Graph-yielding FinSequence such that
    A1: p.1 == H & p.len p = G & len p = G.size() - H.size() + 1 and
    A2: 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 by Th64;
  take p;
  thus p.1 is edgeless & p.len p = G by A1, Th52;
  thus len p = G.size() - 0 + 1 by A1, Th49 .= G.size() + 1;
  thus thesis by A2;
end;
