reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;

theorem Th3:
  for D being finite set, n being Element of NAT, X being set st X
  = {x where x is Element of D* : len x <= n } holds X is finite
proof
  let D be finite set,n be Element of NAT, X be set;
  set B = {x where x is Element of D* : 1 <= len x & len x <= n };
  assume
A1: X = {x where x is Element of D* : len x <= n };
A2: X c= { {} } \/ B
  proof
    let y be object;
    assume y in X;
    then consider x being Element of D* such that
A3: y=x and
A4: len x <= n by A1;
    per cases;
    suppose
      len x < 0+1;
      then x = {} by NAT_1:13;
      then x in { {} } by TARSKI:def 1;
      hence thesis by A3,XBOOLE_0:def 3;
    end;
    suppose
      len x >= 0+1;
      then x in B by A4;
      hence thesis by A3,XBOOLE_0:def 3;
    end;
  end;
  B is finite by Th2;
  hence thesis by A2;
end;
