reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th11:
  for f be real-valued Function st f is bounded_above holds
    rng f is bounded_above
proof
  let f be real-valued Function;
  set X = dom f;
AA: f|X = f;
  assume f is bounded_above;
  then consider a be Real such that
A1: for x1 being object st x1 in X /\ dom f holds f.x1 <= a by AA,RFUNCT_1:70;
  a is UpperBound of rng f
  proof
    let y be ExtReal;
    assume y in rng f;
    then ex s being object st s in dom f & y = f.s by FUNCT_1:def 3;
    hence thesis by A1;
  end;
  hence thesis;
end;
