
theorem Th30:
  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_above & F2 is bounded_above holds F1 + F2 is bounded_above
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_above and
A2: F2 is bounded_above;
A4: sup F1 in REAL & sup F2 in REAL implies sup F1 + sup F2 <+infty
  proof
    reconsider a = sup F1, b = sup F2 as R_eal;
    assume that
A5: sup F1 in REAL and
A6: sup F2 in REAL;
    reconsider a,b as Element of REAL by A5,A6;
    sup F1 + sup F2 = a + b by XXREAL_3:def 2;
    hence thesis by XXREAL_0:9;
  end;
A7: sup F1 in REAL & sup F2 = -infty implies sup F1 + sup F2 <+infty by
XXREAL_0:7,XXREAL_3:def 2;
A8: sup F1 = -infty & sup F2 = -infty implies sup F1 + sup F2 <+infty by
XXREAL_0:7,XXREAL_3:def 2;
A9: sup F1 = -infty & sup F2 in REAL implies sup F1 + sup F2 <+infty by
XXREAL_0:7,XXREAL_3:def 2;
  sup(F1 + F2) <+infty
  proof
    assume not sup(F1 + F2) <+infty;
    then not sup(F1 + F2) <= +infty or sup(F1 + F2) = +infty by XXREAL_0:1;
    hence thesis by A1,A2,A4,A7,A9,A8,Th16,XXREAL_0:4,XXREAL_3:def 1;
  end;
  hence thesis;
end;
