let r be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL
for D being Division of A st f | A is bounded & r <= 0 holds
(lower_sum_set (r (#) f)) . D <= (r * (lower_bound (rng f))) * (vol A)

let A be non empty closed_interval Subset of REAL; :: thesis: for f being Function of A,REAL
for D being Division of A st f | A is bounded & r <= 0 holds
(lower_sum_set (r (#) f)) . D <= (r * (lower_bound (rng f))) * (vol A)

let f be Function of A,REAL; :: thesis: for D being Division of A st f | A is bounded & r <= 0 holds
(lower_sum_set (r (#) f)) . D <= (r * (lower_bound (rng f))) * (vol A)

let D be Division of A; :: thesis: ( f | A is bounded & r <= 0 implies (lower_sum_set (r (#) f)) . D <= (r * (lower_bound (rng f))) * (vol A) )
assume that
A1: f | A is bounded and
A2: r <= 0 ; :: thesis: (lower_sum_set (r (#) f)) . D <= (r * (lower_bound (rng f))) * (vol A)
A3: rng f is bounded_below by ;
A4: (r (#) f) | A is bounded by ;
then A5: upper_sum ((r (#) f),D) <= (upper_bound (rng (r (#) f))) * (vol A) by INTEGRA1:27;
(lower_sum_set (r (#) f)) . D = lower_sum ((r (#) f),D) by INTEGRA1:def 11;
then A6: (lower_sum_set (r (#) f)) . D <= upper_sum ((r (#) f),D) by ;
upper_bound (rng (r (#) f)) = upper_bound (r ** (rng f)) by Th17
.= r * (lower_bound (rng f)) by A2, A3, Th16 ;
hence (lower_sum_set (r (#) f)) . D <= (r * (lower_bound (rng f))) * (vol A) by ; :: thesis: verum