
theorem Th61:
  for G being _finite edgeless _Graph, H being Subgraph of G
  ex p being non empty _finite edgeless Graph-yielding FinSequence
  st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 &
    for n being Element of dom p st n <= len p - 1 holds
    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 edgeless _Graph, H be Subgraph of G;
  G.size() = 0 by Th49
    .= H.size() by Th49;
  then consider p being non empty _finite 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 holds
      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 Th60;
  defpred P[Nat] means for n being Element of dom p st $1 = n
    holds p.n is edgeless;
  A3: P[1] by A1, Th52;
  A4: for k being non zero Nat st P[k] holds P[k+1]
  proof
    let k be non zero Nat;
    assume A5: P[k];
    let m be Element of dom p;
    assume A6: k+1 = m;
    then A7: k+1 <= len p by FINSEQ_3:25;
    then A8: k+1-1 <= len p - 0 by XREAL_1:13;
    1 <= k by NAT_1:14;
    then reconsider n = k as Element of dom p by A8, FINSEQ_3:25;
    k+1-1 <= len p - 1 by A7, XREAL_1:9;
    then consider v being Vertex of G such that
      A9: p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by A2;
    p.n is edgeless by A5;
    hence thesis by A6, A9;
  end;
  A10: for k being non zero Nat holds P[k] from NAT_1:sch 10(A3,A4);
  for x being Element of dom p holds p.x is edgeless
  proof
    let x be Element of dom p;
    x is non zero Nat by FINSEQ_3:25;
    hence thesis by A10;
  end;
  then p is edgeless;
  then reconsider p as non empty _finite edgeless Graph-yielding FinSequence;
  take p;
  thus thesis by A1, A2;
end;
