let r be Real; :: thesis: for A being 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
(upper_sum_set (r (#) f)) . D >= (r * (upper_bound (rng f))) * (vol A)
let A be 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
(upper_sum_set (r (#) f)) . D >= (r * (upper_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
(upper_sum_set (r (#) f)) . D >= (r * (upper_bound (rng f))) * (vol A)
let D be Division of A; :: thesis: ( f | A is bounded & r <= 0 implies (upper_sum_set (r (#) f)) . D >= (r * (upper_bound (rng f))) * (vol A) )
assume that
A1: f | A is bounded and
A2: r <= 0 ; :: thesis: (upper_sum_set (r (#) f)) . D >= (r * (upper_bound (rng f))) * (vol A)
A3: rng f is bounded_above by A1, INTEGRA1:15;
A4: (r (#) f) | A is bounded by A1, RFUNCT_1:97;
then A5: lower_sum (r (#) f),D >= (lower_bound (rng (r (#) f))) * (vol A) by INTEGRA1:27;
( dom (upper_sum_set (r (#) f)) = divs A & D in divs A ) by INTEGRA1:def 3, INTEGRA1:def 11;
then (upper_sum_set (r (#) f)) . D = upper_sum (r (#) f),D by INTEGRA1:def 11;
then A6: (upper_sum_set (r (#) f)) . D >= lower_sum (r (#) f),D by A4, INTEGRA1:30;
lower_bound (rng (r (#) f)) = lower_bound (r ** (rng f)) by Th17
.= r * (upper_bound (rng f)) by A2, A3, Th14 ;
hence (upper_sum_set (r (#) f)) . D >= (r * (upper_bound (rng f))) * (vol A) by A6, A5, XXREAL_0:2; :: thesis: verum