
theorem Th23:
for I,J be non empty closed_interval Subset of REAL, K be Subset of REAL,
  z be Element of REAL,
  f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
  g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
  Pg2 be PartFunc of [:REAL,REAL:],REAL
  st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:]
   & f = g & Pg2 = ProjPMap2(R_EAL g,z) holds
    Pg2 is_integrable_on Prod_Measure(L-Meas,L-Meas)
  & Integral(Prod_Measure(L-Meas,L-Meas),Pg2)
     = Integral(Prod_Measure(L-Meas,L-Meas),(ProjPMap2(R_EAL g,z)))
  & Integral(Prod_Measure(L-Meas,L-Meas),Pg2)
     = (Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g)).z
proof
    let I,J be non empty closed_interval Subset of REAL, K be Subset of REAL,
    z be Element of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    Pg2 be PartFunc of [:REAL,REAL:],REAL;
    assume that
A1: z in K and
A2: dom f = [:[:I,J:],K:] and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: Pg2 = ProjPMap2(R_EAL g,z);

    reconsider Pf2 = Pg2 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;

A6: dom Pf2 = [:I,J:] by A1,A2,A4,A5,MESFUN16:28; then
    Pf2 is_continuous_on [:I,J:] by A2,A3,A4,A5,Th18;
    hence Pg2 is_integrable_on Prod_Measure(L-Meas,L-Meas)
      by A6,MESFUN16:57;

    thus Integral(Prod_Measure(L-Meas,L-Meas),Pg2)
     = Integral(Prod_Measure(L-Meas,L-Meas),(ProjPMap2(R_EAL g,z)))
       by A5,MESFUNC5:def 7;
    R_EAL Pg2 = ProjPMap2(R_EAL g,z) by A5,MESFUNC5:def 7;
    hence Integral(Prod_Measure(L-Meas,L-Meas),Pg2)
     = (Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g)).z by MESFUN12:def 7;
end;
