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 M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )
let M2 be sigma_Measure of S2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )
let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( M1 is sigma_finite & M2 is sigma_finite implies ( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) ) )
assume that
A1: M1 is sigma_finite and
A2: M2 is sigma_finite ; :: thesis: ( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )
( Integral (M2,(X-vol (E,M1))) = (product_sigma_Measure (M1,M2)) . E & Integral (M1,(Y-vol (E,M2))) = (product_sigma_Measure (M1,M2)) . E ) by A1, A2, MEASUR11:118, MEASUR11:117;
hence ( Integral (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) ) by MESFUNC9:14; :: thesis: verum