
theorem
for I,J be non empty closed_interval Subset of REAL,
 f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL
 st [:I,J:] = dom f & f is_continuous_on [:I,J:] & f = g holds
   g is_integrable_on Prod_Measure(L-Meas,L-Meas)
 & (for x being Element of REAL
       holds Integral2(L-Meas,|.R_EAL g.|).x < +infty)
 & (for y being Element of REAL
       holds Integral1(L-Meas,|.R_EAL g.|).y < +infty)
 & (for U being Element of L-Field
       holds Integral2(L-Meas,R_EAL g) is U-measurable)
 & (for V being Element of L-Field
       holds Integral1(L-Meas,R_EAL g) is V-measurable)
 & Integral2(L-Meas,R_EAL g) is_integrable_on L-Meas
 & Integral1(L-Meas,R_EAL g) is_integrable_on L-Meas
 & Integral(Prod_Measure(L-Meas,L-Meas),g)
      = Integral(L-Meas,Integral2(L-Meas,R_EAL g))
 & Integral(Prod_Measure(L-Meas,L-Meas),g)
      = Integral(L-Meas,Integral1(L-Meas,R_EAL g)) by Lm4,Lm5;
