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))
for V being Element of S2
for f being b4 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.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))
for V being Element of S2
for f being b3 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.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))
for V being Element of S2
for f being b2 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable

let M1 be sigma_Measure of S1; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2))
for V being Element of S2
for f being b1 -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable

let E be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for V being Element of S2
for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable

let V be Element of S2; :: thesis: for f being E -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & E = dom f holds
Integral1 (M1,|.f.|) is V -measurable

let f be E -measurable PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( M1 is sigma_finite & E = dom f implies Integral1 (M1,|.f.|) is V -measurable )
assume that
A1: M1 is sigma_finite and
A2: E = dom f ; :: thesis: Integral1 (M1,|.f.|) is V -measurable
A3: E = dom |.f.| by ;
|.f.| is E -measurable by ;
hence Integral1 (M1,|.f.|) is V -measurable by ; :: thesis: verum