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 f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

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 f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1

for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) implies ( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 ) )

assume that

A1: M1 is sigma_finite and

A2: f is_integrable_on Prod_Measure (M1,M2) ; :: thesis: ( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

consider E being Element of sigma (measurable_rectangles (S1,S2)) such that

A3: ( E = dom f & f is E -measurable ) by A2, MESFUNC5:def 17;

reconsider SX2 = X2 as Element of S2 by MEASURE1:7;

A4: Integral1 (M1,|.f.|) is SX2 -measurable by A1, A3, Th5;

( E = dom |.f.| & |.f.| is E -measurable ) by A3, MESFUNC1:def 10, MESFUNC2:27;

then A5: Integral1 (M1,|.f.|) is V97() by MESFUN12:66;

hence Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) by MESFUN11:31; :: thesis: Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0

SX2 = dom (Integral1 (M1,|.f.|)) by FUNCT_2:def 1;

hence Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 by A4, A5, MESFUN11:53; :: thesis: verum

for S2 being SigmaField of X2

for M1 being sigma_Measure of S1

for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

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 f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1

for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2

for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) holds

( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & f is_integrable_on Prod_Measure (M1,M2) implies ( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 ) )

assume that

A1: M1 is sigma_finite and

A2: f is_integrable_on Prod_Measure (M1,M2) ; :: thesis: ( Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) & Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 )

consider E being Element of sigma (measurable_rectangles (S1,S2)) such that

A3: ( E = dom f & f is E -measurable ) by A2, MESFUNC5:def 17;

reconsider SX2 = X2 as Element of S2 by MEASURE1:7;

A4: Integral1 (M1,|.f.|) is SX2 -measurable by A1, A3, Th5;

( E = dom |.f.| & |.f.| is E -measurable ) by A3, MESFUNC1:def 10, MESFUNC2:27;

then A5: Integral1 (M1,|.f.|) is V97() by MESFUN12:66;

hence Integral (M2,(max+ (Integral1 (M1,|.f.|)))) = Integral (M2,(Integral1 (M1,|.f.|))) by MESFUN11:31; :: thesis: Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0

SX2 = dom (Integral1 (M1,|.f.|)) by FUNCT_2:def 1;

hence Integral (M2,(max- (Integral1 (M1,|.f.|)))) = 0 by A4, A5, MESFUN11:53; :: thesis: verum