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
for A being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
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
for A being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
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
for A being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
let M1 be sigma_Measure of S1; :: thesis: for M2 being sigma_Measure of S2
for f being PartFunc of [:X1,X2:],ExtREAL
for A being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
let M2 be sigma_Measure of S2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for A being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for A being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f holds
( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f implies ( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) ) )
assume a1: ( M1 is sigma_finite & M2 is sigma_finite & f is_simple_func_in sigma (measurable_rectangles (S1,S2)) & ( f is nonnegative or f is nonpositive ) & A = dom f ) ; :: thesis: ( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )

per cases ( not f is empty or f is empty ) ;

suppose not f is empty ; :: thesis: ( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )

hence ( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) ) by a1, Lm13; :: thesis: verum

end;

suppose f is empty ; :: thesis: ( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )

hence ( Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) & Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) ) by a1, Lm14; :: thesis: verum

end;

end;