
theorem Th18:
  for G being _finite _Graph, S being VNumberingSeq of G, n being
  Nat holds S.n is one-to-one
proof
  let G being _finite _Graph, S be VNumberingSeq of G, n be Nat;
  defpred P[Nat] means S.$1 is one-to-one;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    set GN = S.Lifespan();
    let k be Nat such that
A2: P[k];
    set w = S.PickedAt(k);
    set CK1 = S.(k+1);
    set CSK = S.k;
    set VLK = CSK;
    set VL1 = CK1;
    per cases;
    suppose
A3:   k < GN;
      set wf = w .--> (GN -' k);
A4:   now
        assume
A5:     GN -' k in rng VLK;
        rng VLK = (Seg GN) \ Seg (GN -' k) by Th14;
        hence contradiction by A5,Th3;
      end;
      rng wf = {GN -' k} by FUNCOP_1:8;
      then
A6:   rng wf misses rng VLK by A4,ZFMISC_1:50;
      VL1 = VLK +* (w .--> (GN -' k)) by A3,Def9;
      hence thesis by A2,A6,FUNCT_4:92;
    end;
    suppose
A7:   k >= GN;
      k <= k+1 by NAT_1:13;
      hence thesis by A2,A7,Th10;
    end;
  end;
A8: P[ 0 ] by Def8;
  for k being Nat holds P[k] from NAT_1:sch 2(A8,A1);
  hence thesis;
end;
