reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;

theorem Th14:
  for P, Q, n st Q c= P* holds Q^^n c= P*
proof
  let P, Q, n;
  assume A1: Q c= P*;
  defpred X[ Nat ] means Q^^$1 c= P*;
  A2: X[ 0 ]
  proof
    Q^^0 = {{}} by Th6 .= P^^0 by Th6;
    hence thesis by Th8;
  end;
  A3: for k holds X[ k ] implies X[ k+1 ]
  proof
    let k;
    assume X[ k ];
    then (Q^^k)^Q c= P* by A1, Th13;
    hence thesis by Th6;
  end;
  for k holds X[ k ] from NAT_1:sch 2(A2, A3);
  hence thesis;
end;
