let X be non empty set ; :: thesis: for Y, Z being non empty Subset of ExtREAL

for F1 being Function of X,Y

for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds

F1 + F2 is bounded_above

let Y, Z be non empty Subset of ExtREAL; :: thesis: for F1 being Function of X,Y

for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds

F1 + F2 is bounded_above

let F1 be Function of X,Y; :: thesis: for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds

F1 + F2 is bounded_above

let F2 be Function of X,Z; :: thesis: ( F1 is bounded_above & F2 is bounded_above implies F1 + F2 is bounded_above )

assume that

A1: F1 is bounded_above and

A2: F2 is bounded_above ; :: thesis: F1 + F2 is bounded_above

A4: ( sup F1 in REAL & sup F2 in REAL implies (sup F1) + (sup F2) < +infty )

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

for F1 being Function of X,Y

for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds

F1 + F2 is bounded_above

let Y, Z be non empty Subset of ExtREAL; :: thesis: for F1 being Function of X,Y

for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds

F1 + F2 is bounded_above

let F1 be Function of X,Y; :: thesis: for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds

F1 + F2 is bounded_above

let F2 be Function of X,Z; :: thesis: ( F1 is bounded_above & F2 is bounded_above implies F1 + F2 is bounded_above )

assume that

A1: F1 is bounded_above and

A2: F2 is bounded_above ; :: thesis: F1 + F2 is bounded_above

A4: ( sup F1 in REAL & sup F2 in REAL implies (sup F1) + (sup F2) < +infty )

proof

A7:
( sup F1 in REAL & sup F2 = -infty implies (sup F1) + (sup F2) < +infty )
by XXREAL_0:7, XXREAL_3:def 2;
reconsider a = sup F1, b = sup F2 as R_eal ;

assume that

A5: sup F1 in REAL and

A6: sup F2 in REAL ; :: thesis: (sup F1) + (sup F2) < +infty

reconsider a = a, b = b as Element of REAL by A5, A6;

(sup F1) + (sup F2) = a + b by XXREAL_3:def 2;

hence (sup F1) + (sup F2) < +infty by XXREAL_0:9; :: thesis: verum

end;assume that

A5: sup F1 in REAL and

A6: sup F2 in REAL ; :: thesis: (sup F1) + (sup F2) < +infty

reconsider a = a, b = b as Element of REAL by A5, A6;

(sup F1) + (sup F2) = a + b by XXREAL_3:def 2;

hence (sup F1) + (sup F2) < +infty by XXREAL_0:9; :: thesis: verum

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

hence
F1 + F2 is bounded_above
; :: thesis: verum
assume
not sup (F1 + F2) < +infty
; :: thesis: contradiction

then ( not sup (F1 + F2) <= +infty or sup (F1 + F2) = +infty ) by XXREAL_0:1;

hence contradiction by A1, A2, A4, A7, A9, A8, Th16, XXREAL_0:4, XXREAL_3:def 1; :: thesis: verum

end;then ( not sup (F1 + F2) <= +infty or sup (F1 + F2) = +infty ) by XXREAL_0:1;

hence contradiction by A1, A2, A4, A7, A9, A8, Th16, XXREAL_0:4, XXREAL_3:def 1; :: thesis: verum