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
  for f be real-valued Function st rng f is bounded_below holds
    f is bounded_below
proof
  let f be real-valued Function;
  set X = dom f;
  assume rng f is bounded_below;
  then consider a be Real such that
A1: a is LowerBound of rng f;
AA: f|X = f;
  for x1 being object st x1 in X /\ dom f holds a <= f.x1
  proof
    let x1 be object;
    assume x1 in X /\ dom f;
    then f.x1 in rng f by FUNCT_1:def 3;
    hence thesis by A1,XXREAL_2:def 2;
  end;
  hence thesis by AA,RFUNCT_1:71;
end;
