theorem :: INTEGRA1:26

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))