
theorem
for x be Element of REAL, 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,
  P1Gz be PartFunc of REAL,REAL st
  x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
& P1Gz = ProjPMap1(Integral2(L-Meas,R_EAL g)| [:I,J:],x)
holds
    P1Gz is continuous
    & dom ProjPMap1(Integral2 (L-Meas,R_EAL g)| [:I,J:],x) = J
    & P1Gz|J is bounded & P1Gz is_integrable_on J
    & integral(P1Gz,J)
       = Integral(L-Meas,(ProjPMap1(Integral2(L-Meas,R_EAL g)| [:I,J:],x)))
    & integral(P1Gz,J)
        = (Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])).x
    & ProjPMap1(Integral2(L-Meas,R_EAL g)| [:I,J:],x) is_integrable_on L-Meas
proof
    let x be Element of REAL,
    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 P1Gz be PartFunc of REAL,REAL;
    assume that
A1:x in I and
A2: [:[:I,J:],K:] = dom f and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: P1Gz = ProjPMap1(Integral2(L-Meas,R_EAL g)| [:I,J:],x);

    reconsider Gz = Integral2(L-Meas,R_EAL g)| [:I,J:]
      as PartFunc of [:REAL,REAL:],REAL by A2,A3,A4,Th32;
    reconsider Fz = Gz as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;

A6: dom Integral2(L-Meas,R_EAL g) = [:REAL,REAL:] by FUNCT_2:def 1;

A7: P1Gz = ProjPMap1(R_EAL Gz,x) by A5,MESFUNC5:def 7;

A8: Fz is_uniformly_continuous_on [:I,J:] by A2,A3,A4,Th34;
    hence P1Gz is continuous by A6,A7,NFCONT_2:7,MESFUN16:36;

    dom ProjPMap1(R_EAL Gz,x) = J by A6,A1,A8,MESFUN16:27;
    hence dom ProjPMap1(Integral2 (L-Meas,R_EAL g)| [:I,J:],x) = J
      by MESFUNC5:def 7;

A9: Fz is_continuous_on [:I,J:] by A8,NFCONT_2:7;
    hence P1Gz|J is bounded & P1Gz is_integrable_on J by A6,A7,A1,MESFUN16:40;

A10: P1Gz is_integrable_on L-Meas
  & integral(P1Gz,J) = Integral(L-Meas,(ProjPMap1(R_EAL Gz,x)))
  & (Integral2(L-Meas,R_EAL Gz)).x = integral(P1Gz,J)
      by A6,A7,A9,A1,MESFUN16:41;
    hence
    integral(P1Gz,J)
      = Integral(L-Meas,ProjPMap1(Integral2(L-Meas,R_EAL g)| [:I,J:],x))
  & integral(P1Gz,J) = (Integral2(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])).x
      by MESFUNC5:def 7;

    ProjPMap1(R_EAL Gz,x) is_integrable_on L-Meas by A10,A7,MESFUNC5:def 7;
    hence ProjPMap1(Integral2(L-Meas,R_EAL g)| [:I,J:],x)
     is_integrable_on L-Meas by MESFUNC5:def 7;
end;
