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 * (lower_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 * (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
(upper_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 (upper_sum_set (r (#) f)) . D >= (r * (lower_bound (rng f))) * (vol A) )
assume that
A1: f | A is bounded and
A2: r >= 0 ; :: thesis: (upper_sum_set (r (#) f)) . D >= (r * (lower_bound (rng f))) * (vol A)
A3: dom (upper_sum_set (r (#) f)) = divs A by INTEGRA1:def 11;
Y: (r (#) f) | A is bounded by A1, RFUNCT_1:97;
then A4: ( (r (#) f) | A is bounded_below & (r (#) f) | A is bounded_above ) ;
( f | A is bounded_below & f | A is bounded_above ) by A1;
then A5: rng f is bounded_below by INTEGRA1:13;
D in divs A by INTEGRA1:def 3;
then (upper_sum_set (r (#) f)) . D = upper_sum (r (#) f),D by A3, INTEGRA1:def 11;
then A6: (upper_sum_set (r (#) f)) . D >= lower_sum (r (#) f),D by Y, INTEGRA1:30;
A7: lower_sum (r (#) f),D >= (lower_bound (rng (r (#) f))) * (vol A) by A4, INTEGRA1:27;
lower_bound (rng (r (#) f)) = lower_bound (r ** (rng f)) by Th17
.= r * (lower_bound (rng f)) by A2, A5, Th15 ;
hence (upper_sum_set (r (#) f)) . D >= (r * (lower_bound (rng f))) * (vol A) by A6, A7, XXREAL_0:2; :: thesis: verum