
theorem Th28:
  for X being non empty set, Y being non empty Subset of ExtREAL,
  F being Function of X,Y st Y c= REAL holds (F is bounded_below iff inf F in
  REAL)
proof
  let X be non empty set, Y be non empty Subset of ExtREAL, F be Function of X
  ,Y;
  assume
A1: Y c= REAL;
  thus F is bounded_below implies inf F in REAL
  proof
    set x = the Element of X;
    X = dom F by FUNCT_2:def 1;
    then
A2: F.x in rng F by FUNCT_1:def 3;
    rng F c= Y by RELAT_1:def 19;
    then F.x in Y by A2;
    then
A3: not inf F = +infty by A1,Th26,XXREAL_0:9;
    assume a4: F is bounded_below;
    inf F in REAL or inf F in {-infty,+infty} by XBOOLE_0:def 3;
    hence thesis by a4,A3,TARSKI:def 2;
  end;
  assume inf F in REAL;
  hence thesis by XXREAL_0:12;
end;
