
theorem
  for G being _finite _Graph, S being VNumberingSeq of G, n being Nat st
  n <= S.Lifespan() holds card dom (S.n) = n
proof
  let G be _finite _Graph, S be VNumberingSeq of G, n be Nat such that
A1: n <= S.Lifespan();
  defpred P[Nat] means $1 <= S.Lifespan() implies card dom (S.$1) = $1;
A2: for k being Nat st k < S.Lifespan() & card dom (S.k) = k holds card dom
  (S.(k+1)) = k+1
  proof
    let k be Nat such that
A3: k < S.Lifespan() and
A4: card dom (S.k) = k;
    set w = S.PickedAt(k);
    set CK1 = S.(k+1);
    set CSK = S.k;
    set VLK = CSK;
    set VK1 = CK1;
    set wf = w .--> (S.Lifespan() -' k);
A5: dom wf = {w};
    VK1 = VLK +* (w .--> (S.Lifespan()-'k)) by A3,Def9;
    then
A6: dom VK1 = dom VLK \/ {w} by A5,FUNCT_4:def 1;
    not w in dom VLK by A3,Def9;
    hence thesis by A4,A6,CARD_2:41;
  end;
A7: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A8: P[k];
    per cases;
    suppose
      k < S.Lifespan();
      hence thesis by A2,A8;
    end;
    suppose
      k >= S.Lifespan();
      hence thesis by NAT_1:13;
    end;
  end;
A9: P[ 0 ] by Def8,CARD_1:27,RELAT_1:38;
  for k being Nat holds P[k] from NAT_1:sch 2(A9,A7);
  hence thesis by A1;
end;
