
theorem
for y 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,
  P2Gz be PartFunc of REAL,REAL st
  y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
& P2Gz = ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y)
holds
    P2Gz is continuous
    & dom ProjPMap2(Integral2 (L-Meas,R_EAL g)| [:I,J:],y) = I
    & P2Gz|I is bounded & P2Gz is_integrable_on I
    & integral(P2Gz,I)
       = Integral(L-Meas,(ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y)))
    & integral(P2Gz,I)
        = (Integral1(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])).y
    & ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y) is_integrable_on L-Meas
proof
    let y 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 P2Gz be PartFunc of REAL,REAL;
    assume that
A1: y in J and
A2: [:[:I,J:],K:] = dom f and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: P2Gz = ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y);

    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: P2Gz = ProjPMap2(R_EAL Gz,y) by A5,MESFUNC5:def 7;

A8: Fz is_uniformly_continuous_on [:I,J:] by A2,A3,A4,Th34;
    hence P2Gz is continuous by A6,A7,NFCONT_2:7,MESFUN16:37;

    dom ProjPMap2(R_EAL Gz,y) = I by A6,A1,A8,MESFUN16:28;
    hence dom ProjPMap2(Integral2 (L-Meas,R_EAL g)| [:I,J:],y) = I
      by MESFUNC5:def 7;

A9: Fz is_continuous_on [:I,J:] by A8,NFCONT_2:7;
    hence P2Gz|I is bounded & P2Gz is_integrable_on I by A6,A7,A1,MESFUN16:42;

A10: P2Gz is_integrable_on L-Meas
  & integral(P2Gz,I) = Integral(L-Meas,(ProjPMap2(R_EAL Gz,y)))
  & (Integral1(L-Meas,R_EAL Gz)).y = integral(P2Gz,I)
      by A6,A7,A9,A1,MESFUN16:43;
    hence
    integral(P2Gz,I)
      = Integral(L-Meas,ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y))
  & integral(P2Gz,I) = (Integral1(L-Meas,Integral2(L-Meas,R_EAL g)| [:I,J:])).y
      by MESFUNC5:def 7;

    ProjPMap2(R_EAL Gz,y) is_integrable_on L-Meas by A10,A7,MESFUNC5:def 7;
    hence ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y)
     is_integrable_on L-Meas by MESFUNC5:def 7;
end;
