
theorem
  for p being non empty Graph-yielding FinSequence
  st p.1 is edgeless &
    for n being Element of dom p st n <= len p - 1 holds
    ex v being object st p.(n+1) is addVertex of p.n,v
  holds p.len p is edgeless
proof
  defpred P[Nat] means for p being non empty Graph-yielding FinSequence
    st len p = $1 & p.1 is edgeless &
    for n being Element of dom p st n <= len p - 1 holds
      ex v being object st p.(n+1) is addVertex of p.n,v
    holds p.len p is edgeless;
  A1: P[1];
  A2: for m being non zero Nat st P[m] holds P[m+1]
  proof
    let m be non zero Nat;
    assume A3: P[m];
    let p be non empty Graph-yielding FinSequence;
    assume that
      A4: len p = m + 1 & p.1 is edgeless and
      A5: for n being Element of dom p st n <= len p - 1 holds
        ex v being object st p.(n+1) is addVertex of p.n,v;
    reconsider q = p|m as non empty Graph-yielding FinSequence;
    m+1-1 <= len p - 0 by A4, XREAL_1:10;
    then A6: len q = m by FINSEQ_1:59;
    A7: 1 <= m by INT_1:74;
    then A8: q.1 is edgeless by A4, FINSEQ_3:112;
    now
      let n be Element of dom q;
      assume A9: n <= len q - 1;
      then n + 0 <= m - 1 + 1 by A6, XREAL_1:7;
      then A10: n <= len p - 1 by A4;
      A11: 1 <= n & n <= len q by FINSEQ_3:25;
      then 1 <= n & n + 0 <= m + 1 by A6, XREAL_1:7;
      then reconsider k = n as Element of dom p by A4, FINSEQ_3:25;
      consider v being object such that
        A12: p.(k+1) is addVertex of p.k,v by A5, A10;
      take v;
      n+1 <= len q - 1 + 1 by A9, XREAL_1:6;
      then p.(k+1) = q.(n+1) & p.k = q.k by A6, A11, FINSEQ_3:112;
      hence q.(n+1) is addVertex of q.n,v by A12;
    end;
    then A13: q.len q is edgeless by A3, A6, A8;
    m+0 <= len p - 1 + 1 by A4, XREAL_1:6;
    then reconsider k=m as Element of dom p by A7, FINSEQ_3:25;
    consider v be object such that
      A14: p.(k+1) is addVertex of p.k,v by A5, A4;
    p.k = q.len q by A6, FINSEQ_3:112;
    hence thesis by A13, A4, A14;
  end;
  A15: for m being non zero Nat holds P[m] from NAT_1:sch 10(A1,A2);
  let p be non empty Graph-yielding FinSequence;
  assume that
    A16: p.1 is edgeless &
      for n being Element of dom p st n <= len p - 1 holds
        ex v being object st p.(n+1) is addVertex of p.n,v;
  thus thesis by A15, A16;
end;
