let i be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i))
let A be closed-interval Subset of REAL ; :: thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i))
let D be Division of A; :: thesis: for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i))
let f be Function of A,REAL ; :: thesis: ( f | A is bounded_above & i in dom D implies (upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i)) )
A1: dom f = A by FUNCT_2:def 1;
assume f | A is bounded_above ; :: thesis: ( not i in dom D or (upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i)) )
then A2: rng f is bounded_above by Th15;
assume i in dom D ; :: thesis: (upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i))
then dom (f | (divset D,i)) = divset D,i by A1, Th10, RELAT_1:91;
then A3: rng (f | (divset D,i)) is non empty Subset of REAL by RELAT_1:65;
A4: 0 <= vol (divset D,i) by SEQ_4:24, XREAL_1:50;
rng (f | (divset D,i)) c= rng f by RELAT_1:99;
hence (upper_bound (rng f)) * (vol (divset D,i)) >= (upper_bound (rng (f | (divset D,i)))) * (vol (divset D,i)) by A3, A2, A4, SEQ_4:65, XREAL_1:66; :: thesis: verum