let A be non empty closed_interval Subset of REAL; :: thesis: for f being Function of A,REAL
for T being DivSequence of A
for S being middle_volume_Sequence of f,T
for i being Element of NAT st f | A is bounded_above holds
(middle_sum (f,S)) . i <= (upper_sum (f,T)) . i

let f be Function of A,REAL; :: thesis: for T being DivSequence of A
for S being middle_volume_Sequence of f,T
for i being Element of NAT st f | A is bounded_above holds
(middle_sum (f,S)) . i <= (upper_sum (f,T)) . i

let T be DivSequence of A; :: thesis: for S being middle_volume_Sequence of f,T
for i being Element of NAT st f | A is bounded_above holds
(middle_sum (f,S)) . i <= (upper_sum (f,T)) . i

let S be middle_volume_Sequence of f,T; :: thesis: for i being Element of NAT st f | A is bounded_above holds
(middle_sum (f,S)) . i <= (upper_sum (f,T)) . i

let i be Element of NAT ; :: thesis: ( f | A is bounded_above implies (middle_sum (f,S)) . i <= (upper_sum (f,T)) . i )
assume A1: f | A is bounded_above ; :: thesis: (middle_sum (f,S)) . i <= (upper_sum (f,T)) . i
( (middle_sum (f,S)) . i = middle_sum (f,(S . i)) & (upper_sum (f,T)) . i = upper_sum (f,(T . i)) ) by ;
hence (middle_sum (f,S)) . i <= (upper_sum (f,T)) . i by ; :: thesis: verum