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 B being Element of S1 holds Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for B being Element of S1 holds Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for B being Element of S1 holds Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let M1 be sigma_Measure of S1; :: thesis: for B being Element of S1 holds Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let B be Element of S1; :: thesis: Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }

now :: thesis: for E being set st E in Field_generated_by (measurable_rectangles (S1,S2)) holds
E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for x being Element of X2 holds F . x = M1 . ((Measurable-Y-section (E,x)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }

let E be set ; :: thesis: ( E in Field_generated_by (measurable_rectangles (S1,S2)) implies E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for x being Element of X2 holds F . x = M1 . ((Measurable-Y-section (E,x)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } )
assume A1: E in Field_generated_by (measurable_rectangles (S1,S2)) ; :: thesis: E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for x being Element of X2 holds F . x = M1 . ((Measurable-Y-section (E,x)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
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 A1;
E1 in Field_generated_by (measurable_rectangles (S1,S2)) by A1;
hence E in { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for x being Element of X2 holds F . x = M1 . ((Measurable-Y-section (E,x)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } by Th79; :: thesis: verum

end;

hence Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } ; :: thesis: verum