let X1, X2 be non empty set ; for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
let S1 be SigmaField of X1; for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
let S2 be SigmaField of X2; for M1 being sigma_Measure of S1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
let M1 be sigma_Measure of S1; for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
let y be Element of X2; for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
let E be Element of sigma (measurable_rectangles (S1,S2)); for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
let r be Real; ( M1 is sigma_finite implies ( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) ) )
assume A1:
M1 is sigma_finite
; ( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )
set p2 = ProjPMap2 ((chi (E,[:X1,X2:])),y);
chi (r,E,[:X1,X2:]) = r (#) (chi (E,[:X1,X2:]))
by Th1;
then A2:
ProjPMap2 ((chi (r,E,[:X1,X2:])),y) = r (#) (ProjPMap2 ((chi (E,[:X1,X2:])),y))
by Th29;
A3:
ProjPMap2 ((chi (E,[:X1,X2:])),y) is nonnegative
by Th32;
A4:
dom (r (#) (X-vol (E,M1))) = X2
by FUNCT_2:def 1;
A5:
chi (E,[:X1,X2:]) is_simple_func_in sigma (measurable_rectangles (S1,S2))
by Th12;
then Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) =
r * (integral' (M1,(ProjPMap2 ((chi (E,[:X1,X2:])),y))))
by A2, A3, Th31, MESFUN11:59
.=
r * ((X-vol (E,M1)) . y)
by A1, Th53
;
hence A7:
(r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y)))
by A4, MESFUNC1:def 6; ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) )
thus
( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) )
verumproof
assume
r >= 0
;
(r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y)))
then A8:
r (#) (ProjPMap2 ((chi (E,[:X1,X2:])),y)) is
nonnegative
by A3, MESFUNC5:20;
r (#) (ProjPMap2 ((chi (E,[:X1,X2:])),y)) is_simple_func_in S1
by A5, Th31, MESFUNC5:39;
hence
(r (#) (X-vol (E,M1))) . y = integral+ (
M1,
(ProjPMap2 ((chi (r,E,[:X1,X2:])),y)))
by A2, A7, A8, MESFUNC5:89;
verum
end;