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 E being Element of sigma (measurable_rectangles (S1,S2)) st E in Field_generated_by (measurable_rectangles (S1,S2)) holds
for B being Element of S1 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 S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2)) st E in Field_generated_by (measurable_rectangles (S1,S2)) holds
for B being Element of S1 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 S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for E being Element of sigma (measurable_rectangles (S1,S2)) st E in Field_generated_by (measurable_rectangles (S1,S2)) holds
for B being Element of S1 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 M1 be sigma_Measure of S1; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st E in Field_generated_by (measurable_rectangles (S1,S2)) holds
for B being Element of S1 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 Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( E in Field_generated_by (measurable_rectangles (S1,S2)) implies for B being Element of S1 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 ) ) } )
assume A0: E in Field_generated_by (measurable_rectangles (S1,S2)) ; :: thesis: for B being Element of S1 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 B be Element of S1; :: 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 ) ) }
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 A0, Th77;
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 ) ) } ; :: thesis: verum