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 V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
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 V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let M2 be sigma_Measure of S2; :: thesis: for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let V be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let A be Element of S1; :: thesis: for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let B be Element of S2; :: thesis: ( M2 is sigma_finite & V = [:A,B:] implies Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } )
assume A1: ( M2 is sigma_finite & V = [:A,B:] ) ; :: thesis: Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let E be object ; :: according to TARSKI:def 3 :: thesis: ( not E in Field_generated_by (measurable_rectangles (S1,S2)) or E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } )
assume A2: E in Field_generated_by (measurable_rectangles (S1,S2)) ; :: thesis: E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
sigma (measurable_rectangles (S1,S2)) = sigma (DisUnion (measurable_rectangles (S1,S2))) by Th1
.= sigma (Field_generated_by (measurable_rectangles (S1,S2))) by SRINGS_3:22 ;
then Field_generated_by (measurable_rectangles (S1,S2)) c= sigma (measurable_rectangles (S1,S2)) by PROB_1:def 9;
then reconsider E1 = E as Element of sigma (measurable_rectangles (S1,S2)) by A2;
E1 in Field_generated_by (measurable_rectangles (S1,S2)) by A2;
hence E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } by A1, Th104; :: thesis: verum