let r be Real; :: thesis: for i being Element of 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_above & r >= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((upper_volume (f,D)) . i)
let i be Element of 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_above & r >= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((upper_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_above & r >= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((upper_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_above & r >= 0 holds
(upper_volume ((r (#) f),D)) . i = r * ((upper_volume (f,D)) . i)
let D be Division of A; :: thesis: ( i in dom D & f | A is bounded_above & r >= 0 implies (upper_volume ((r (#) f),D)) . i = r * ((upper_volume (f,D)) . i) )
assume that
A1: i in dom D and
A2: f | A is bounded_above and
A3: r >= 0 ; :: thesis: (upper_volume ((r (#) f),D)) . i = r * ((upper_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 A1, INTEGRA1:8, XBOOLE_1:28 ;
then A4: not rng (f | (divset (D,i))) is empty by RELAT_1:42;
rng f is bounded_above by A2, INTEGRA1:13;
then A5: rng (f | (divset (D,i))) is bounded_above by RELAT_1:70, XXREAL_2:43;
(upper_volume ((r (#) f),D)) . i = (upper_bound (rng ((r (#) f) | (divset (D,i))))) * (vol (divset (D,i))) by A1, INTEGRA1:def 6
.= (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 * (upper_bound (rng (f | (divset (D,i)))))) * (vol (divset (D,i))) by A3, A4, A5, Th13
.= r * ((upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))))
.= r * ((upper_volume (f,D)) . i) by A1, INTEGRA1:def 6 ;
hence (upper_volume ((r (#) f),D)) . i = r * ((upper_volume (f,D)) . i) ; :: thesis: verum