
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 U being Element of L-Field
      holds Integral2(L-Meas,Integral2(L-Meas,R_EAL g)) is U-measurable)
 & Integral2(L-Meas,Integral2(L-Meas,R_EAL g)) is_integrable_on L-Meas
 & Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g))
     = Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)))
 & Integral(Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas),g)
     = Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)))
 & Integral2(L-Meas,R_EAL g)| [:I,J:]
      is_integrable_on Prod_Measure(L-Meas,L-Meas)
 & Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g)| [:I,J:])
   = Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:]) )
proof
    let I,J,K be non empty closed_interval Subset of REAL;
    let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real;
    let 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;

    set RF = Integral2(L-Meas,R_EAL g);

A4: dom RF = [:REAL,REAL:] by FUNCT_2:def 1;

    reconsider RG = RF| [:I,J:] as PartFunc of [:REAL,REAL:],REAL
      by A1,A2,A3,Th32;
    reconsider RGG = RG as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;

A5:Integral2(L-Meas,R_EAL g) is_integrable_on Prod_Measure(L-Meas,L-Meas)
      by A1,A2,A3,Th43;

A6:for x being Element of REAL holds
      Integral2(L-Meas,|.Integral2(L-Meas,R_EAL g).|).x < +infty
        by A1,A2,A3,Th46;
    hence
    (for U being Element of L-Field
      holds Integral2(L-Meas,Integral2(L-Meas,R_EAL g)) is U-measurable)
  & Integral2(L-Meas,Integral2(L-Meas,R_EAL g)) is_integrable_on L-Meas
  & Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g))
     = Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)))
        by A5,MESFUN13:32,MESFUN16:5;

A7:g is_integrable_on Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas)
      by A1,A2,A3,Th43;

    for x being Element of [:REAL,REAL:] holds
     Integral2(L-Meas,|.R_EAL g.|).x < +infty by A1,A2,A3,Th40; then
    Integral(Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas),R_EAL g)
     = Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g))
       by A7,MESFUN13:32,MESFUN16:5,6;
    hence Integral(Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas),g)
     = Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)))
        by A5,A6,MESFUN13:32,MESFUN16:5;

A8:Integral2(L-Meas,R_EAL g)| [:I,J:] = R_EAL RG by MESFUNC5:def 7;

    RGG is_uniformly_continuous_on [:I,J:] by A1,A2,A3,Th34; then
    RGG is_continuous_on [:I,J:] by NFCONT_2:7; then
    RG is_integrable_on Prod_Measure(L-Meas,L-Meas)
  & Integral(Prod_Measure(L-Meas,L-Meas),RG)
     = Integral(L-Meas,Integral2(L-Meas,R_EAL RG)) by A4,MESFUN16:57;
    hence Integral2(L-Meas,R_EAL g)| [:I,J:]
      is_integrable_on Prod_Measure(L-Meas,L-Meas)
    & Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g)| [:I,J:])
     = Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:]))
        by A8;
end;
