theorem Th1: :: INTEGRA7:1

for a being Real
for A being non empty set
for f, g being Function of A,REAL st rng f is bounded_above & rng g is bounded_above & ( for x being set st x in A holds
|.((f . x) - (g . x)).| <= a ) holds
( (upper_bound (rng f)) - (upper_bound (rng g)) <= a & (upper_bound (rng g)) - (upper_bound (rng f)) <= a )