let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let S2 be SigmaField of X2; :: thesis: for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let M2 be sigma_Measure of S2; :: thesis: for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let x be Element of X1; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let r be Real; :: thesis: ( M2 is sigma_finite implies ( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) ) )
assume A1: M2 is sigma_finite ; :: thesis: ( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
set p2 = ProjPMap1 ((chi (E,[:X1,X2:])),x);
chi (r,E,[:X1,X2:]) = r (#) (chi (E,[:X1,X2:])) by Th1;
then A2: ProjPMap1 ((chi (r,E,[:X1,X2:])),x) = r (#) (ProjPMap1 ((chi (E,[:X1,X2:])),x)) by Th29;
A3: ProjPMap1 ((chi (E,[:X1,X2:])),x) is nonnegative by Th32;
A4: dom (r (#) (Y-vol (E,M2))) = X1 by FUNCT_2:def 1;
A5: chi (E,[:X1,X2:]) is_simple_func_in sigma (measurable_rectangles (S1,S2)) by Th12;
then Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) = r * (integral' (M2,(ProjPMap1 ((chi (E,[:X1,X2:])),x)))) by A2, A3, Th31, MESFUN11:59
.= r * ((Y-vol (E,M2)) . x) by A1, Th52 ;
hence A7: (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) by A4, MESFUNC1:def 6; :: thesis: ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) )
thus ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) :: thesis: verum

proof

assume r >= 0 ; :: thesis: (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x)))
then A8: r (#) (ProjPMap1 ((chi (E,[:X1,X2:])),x)) is nonnegative by A3, MESFUNC5:20;
r (#) (ProjPMap1 ((chi (E,[:X1,X2:])),x)) is_simple_func_in S2 by A5, Th31, MESFUNC5:39;
hence (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) by A2, A7, A8, MESFUNC5:89; :: thesis: verum

end;