
theorem
for I,J,K be non empty closed_interval Subset of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL
 st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
 holds
  (for z being Element of REAL
     holds ProjPMap2(R_EAL g,z) is_integrable_on Prod_Measure(L-Meas,L-Meas))
& (for V being Element of L-Field
     holds Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g) is V-measurable)
& Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g) is_integrable_on L-Meas
& Integral(Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas),g)
    = Integral(L-Meas,Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g))
proof
    let I,J,K be non empty closed_interval Subset of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;

A4:g is_integrable_on Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas)
      by A1,A2,A3,Th43;

    for z being Element of REAL holds
     Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).z < +infty
       by A1,A2,A3,Th41;
    hence
     (for z being Element of REAL holds
        ProjPMap2(R_EAL g,z) is_integrable_on Prod_Measure(L-Meas,L-Meas))
   & (for V being Element of L-Field
       holds Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g) is V-measurable)
   & Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g) is_integrable_on L-Meas
   & Integral(Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas),g)
      = Integral(L-Meas,Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g))
     by A4,MESFUN13:33,MESFUN16:5,6;
end;
