reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;

theorem Th12:
  rng f c= [.a,b.] implies f is bounded
  proof
    assume
A1: rng f c= [.a,b.];
    [.a,b.] c= [.a, +infty .[ & [.a,b.] c= ]. -infty,b.] by XXREAL_1:251,265;
    then rng f c= [.a, +infty .[ & rng f c= ]. -infty,b.] by A1;
    then rng f is bounded_below & rng f is bounded_above by XXREAL_2:43,44;
    then f is bounded_below & f is bounded_above by INTEGRA1:12,14;
    hence thesis;
  end;
