reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;

theorem Th16:
  rng f is real-bounded implies f is real-valued
proof
  assume
A1: rng f is real-bounded;
  then rng f is bounded_above by XXREAL_2:def 11;
  then consider UB being Real such that
A2:  UB is UpperBound of rng f by XXREAL_2:def 10;
A3: UB in REAL by XREAL_0:def 1;
  rng f is bounded_below by A1,XXREAL_2:def 11;
  then consider LB being Real such that
A4: LB is LowerBound of rng f by XXREAL_2:def 9;
A5: LB in REAL by XREAL_0:def 1;
  now
    let x be Element of X;
    assume x in dom f;
    then
A6:  f.x in rng f by FUNCT_1:3;
    then LB <= f.x by A4,XXREAL_2:def 2;
    then -infty < f.x by A5,XXREAL_0:2,12;
    then
A7: -(+infty) < f.x by XXREAL_3:23;
    f.x <= UB by A2,A6,XXREAL_2:def 1;
    then f.x < +infty by A3,XXREAL_0:2,9;
    hence |. f.x .| < +infty by A7,EXTREAL1:22;
  end;
  hence thesis by MESFUNC2:def 1;
end;
