reserve i,j,k,n,n1,n2,m for Nat;
reserve a,r,x,y for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve C for non empty set;
reserve X for set;

theorem Th19:
  for f being PartFunc of A,REAL st f|A is bounded_below holds rng
  (f|X) is bounded_below
proof
  let f be PartFunc of A,REAL;
  assume f|A is bounded_below;
  then rng f is bounded_below by INTEGRA1:11;
  hence thesis by RELAT_1:70,XXREAL_2:44;
end;
