
theorem Th31:
  for X being non empty set, Y,Z being non empty Subset of ExtREAL
  holds for F1 being Function of X,Y, F2 being Function of X,Z st F1 is
  bounded_below & F2 is bounded_below holds F1 + F2 is bounded_below
proof
  let X be non empty set, Y,Z be non empty Subset of ExtREAL;
  let F1 be Function of X,Y, F2 be Function of X,Z;
  assume that
A1: F1 is bounded_below and
A2: F2 is bounded_below;
A4: inf F1 in REAL & inf F2 in REAL implies -infty <inf F1 + inf F2
  proof
    reconsider a = inf F1, b = inf F2 as R_eal;
    assume that
A5: inf F1 in REAL and
A6: inf F2 in REAL;
    reconsider a,b as Element of REAL by A5,A6;
    inf F1 + inf F2 = a + b by XXREAL_3:def 2;
    hence thesis by XXREAL_0:12;
  end;
A7: inf F1 in REAL & inf F2 = +infty implies -infty <inf F1 + inf F2 by
XXREAL_0:7,XXREAL_3:def 2;
A8: inf F1 = +infty & inf F2 = +infty implies -infty <inf F1 + inf F2 by
XXREAL_0:7,XXREAL_3:def 2;
A9: inf F1 = +infty & inf F2 in REAL implies -infty <inf F1 + inf F2 by
XXREAL_0:7,XXREAL_3:def 2;
  -infty <inf(F1 + F2)
  proof
    assume inf(F1 + F2) <= -infty;
    then not -infty <= inf(F1 + F2) or inf(F1 + F2) = -infty by XXREAL_0:1;
    hence thesis by A1,A2,A4,A7,A9,A8,Th17,XXREAL_0:6,XXREAL_3:def 1;
  end;
  hence thesis;
end;
