
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,
  Gzy be PartFunc of REAL,REAL
st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
 & Gzy = Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])|I
holds
    dom Gzy = I
    & Gzy is continuous
    & Gzy|| I is bounded & Gzy is_integrable_on I
    & Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])| I
        is_integrable_on L-Meas
    & Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])| I)
        = integral(Gzy,I)
    & Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g)| [:I,J:])
        = integral(Gzy,I)
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;
    let Gzy be PartFunc of REAL,REAL;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g and
A4: Gzy = Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])|I;

A5: dom Integral2(L-Meas,R_EAL g) = [:REAL,REAL:] by FUNCT_2:def 1;

    reconsider Gz = Integral2(L-Meas,R_EAL g)| [:I,J:]
      as PartFunc of [:REAL,REAL:],REAL by A1,A2,A3,Th32;
    reconsider Fz = Gz as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;

A6: Fz is_uniformly_continuous_on [:I,J:] by A1,A2,A3,Th34; then
A7: Fz is_continuous_on [:I,J:] by NFCONT_2:7;

A8: Gzy = Integral2(L-Meas,R_EAL Gz)|I by A4,MESFUNC5:def 7;

A9:dom(Integral2(L-Meas,R_EAL Gz)) = REAL by FUNCT_2:def 1;
    hence
A10:dom Gzy = I by A8; then
A11:Gzy||I is bounded & Gzy is_integrable_on I
  & Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Gz) = integral(Gzy,I)
      by A7,A5,A8,MESFUN16:53,58,INTEGRA5:10,11;

    thus
    Gzy is continuous by A6,A5,A8,NFCONT_2:7,MESFUN16:53;
    hence Gzy||I is bounded & Gzy is_integrable_on I by A9,A8,INTEGRA5:10,11;

    I in L-Field & J in L-Field by MEASUR10:5,MEASUR12:75; then
    Gzy is_integrable_on L-Meas & Integral(L-Meas,Gzy|I) = integral(Gzy,I)
      by A10,A11,MESFUN14:49;
    hence
    Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])|I
      is_integrable_on L-Meas
  & Integral(L-Meas,Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])|I)
      = integral(Gzy,I) by A4,MESFUNC5:def 7;

    thus
    Integral(Prod_Measure(L-Meas,L-Meas),Integral2(L-Meas,R_EAL g)| [:I,J:])
      = integral(Gzy,I) by A11,MESFUNC5:def 7;
end;
