
theorem Th14:
  for G being _finite _Graph, S being VNumberingSeq of G, n being
  Nat holds rng (S.n) = (Seg S.Lifespan()) \ Seg (S.Lifespan()-'n)
proof
  let G be _finite _Graph, S be VNumberingSeq of G, n be Nat;
  set CSN = S.n;
  set CSO = S.S.Lifespan();
  set GN = S.Lifespan();
  defpred P[Nat] means $1 <= GN implies rng (S.$1) = (Seg GN) \ Seg (GN-'$1);
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A2: P[k];
    set CK1 = S.(k+1);
    set CSK = S.k;
    set VLK = CSK;
    set VK1 = CK1;
    per cases;
    suppose
A3:   k+1 <= GN;
      set w = S.PickedAt(k);
      set wf = w .--> (GN -' k);
A5:   k < GN by A3,NAT_1:13;
      then not w in dom VLK by Def9;
      then
A6:   dom wf misses dom VLK by ZFMISC_1:50;
A7:   rng wf = {GN -' k} by FUNCOP_1:8;
      VK1 = VLK +* (w .--> (GN -' k)) by A5,Def9;
      then rng VK1 = rng VLK \/ {GN -' k} by A7,A6,NECKLACE:6;
      hence thesis by A2,A5,Th5;
    end;
    suppose
      GN < k+1;
      hence thesis;
    end;
  end;
A8: P[ 0 ]
  proof
    set CS0 = S.0;
    set VL0 = CS0;
A9: GN -' 0 = GN - 0 by XREAL_1:233;
    rng VL0 = {} by Def8,RELAT_1:38;
    hence thesis by A9,XBOOLE_1:37;
  end;
A10: for k being Nat holds P[k] from NAT_1:sch 2(A8,A1);
  per cases;
  suppose
    n <= GN;
    hence thesis by A10;
  end;
  suppose
A11: GN < n;
    then GN - n < n - n by XREAL_1:9;
    then GN -' n = 0 by XREAL_0:def 2;
    then
A12: GN -' GN = GN -' n by XREAL_1:232;
    CSO = CSN by A11,Th9;
    hence thesis by A10,A12;
  end;
end;
