let X1, X2 be non empty set ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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)); :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( (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; :: thesis: ( 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))) ) :: thesis: verum

proof

assume r >= 0 ; :: thesis: (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; :: thesis: verum

end;