theorem :: INTEGR15:12

for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f being Function of A,(REAL n) st f is bounded holds
( f is integrable iff ex I being Element of REAL n st
for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = I ) )