reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

theorem
  (omega)*`(card n) c= omega & (card n)*`(omega) c= omega
proof
  defpred P[Nat] means (omega)*`(card $1) c= omega;
A1: P[0];
A2: for k being Nat holds P[k] implies P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    card (k+1) = Segm(k+1)
      .= succ Segm k by NAT_1:38;
    then card (k+1) = card succ k;
    then (omega)*`(card (k+1)) =
    card (succ k *^ omega) by Th13,CARD_1:47
      .= card ( k *^ omega +^ omega) by ORDINAL2:36
      .= card ( k *^ omega) +` omega by Th12,CARD_1:47
      .= (omega)*`(card k) +` omega by Th13,CARD_1:47
      .= omega by A3,Th75;
    hence thesis;
  end;
A4: for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
  hence (omega)*`(card n) c= omega;
  thus thesis by A4;
end;
