let r be Real; :: thesis: for i being Nat
for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL
for D being Division of A st i in dom D & f | A is bounded_below & r <= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i)

let i be Nat; :: 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 i in dom D & f | A is bounded_below & r <= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i)

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 i in dom D & f | A is bounded_below & r <= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i)

let f be Function of A,REAL; :: thesis: for D being Division of A st i in dom D & f | A is bounded_below & r <= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i)

let D be Division of A; :: thesis: ( i in dom D & f | A is bounded_below & r <= 0 implies (upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i) )
assume that
A1: i in dom D and
A2: f | A is bounded_below and
A3: r <= 0 ; :: thesis: (upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i)
dom (f | (divset (D,i))) = (dom f) /\ (divset (D,i)) by RELAT_1:61
.= A /\ (divset (D,i)) by FUNCT_2:def 1
.= divset (D,i) by ;
then A4: not rng (f | (divset (D,i))) is empty by RELAT_1:42;
rng f is bounded_below by ;
then A5: rng (f | (divset (D,i))) is bounded_below by ;
(upper_volume ((r (#) f),D)) . i = (upper_bound (rng ((r (#) f) | (divset (D,i))))) * (vol (divset (D,i))) by
.= (upper_bound (rng (r (#) (f | (divset (D,i)))))) * (vol (divset (D,i))) by RFUNCT_1:49
.= (upper_bound (r ** (rng (f | (divset (D,i)))))) * (vol (divset (D,i))) by Th18
.= (r * (lower_bound (rng (f | (divset (D,i)))))) * (vol (divset (D,i))) by A3, A4, A5, Th16
.= r * ((lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))))
.= r * ((lower_volume (f,D)) . i) by ;
hence (upper_volume ((r (#) f),D)) . i = r * ((lower_volume (f,D)) . i) ; :: thesis: verum