
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
  |.Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).| is Function of REAL,REAL
proof
    let 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;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;
    set Fxy = Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g);
A4: Fxy is Function of REAL,REAL by A1,A2,A3,Th35;
    dom Fxy = REAL by FUNCT_2:def 1; then
A5: dom |.Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).| = REAL
      by MESFUNC1:def 10;
    now let r be object;
     assume
     r in REAL; then
     reconsider r1=r as Element of REAL;
A6:  |.Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).| .r
      = |. Fxy.r1 .| by A5,MESFUNC1:def 10;
     reconsider y = Fxy.r1 as Element of REAL by A4,FUNCT_2:5;
     |.Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).| .r = |. y .|
       by A6,EXTREAL1:12;
     hence |.Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).| .r in REAL
       by XREAL_0:def 1;
    end;
    hence thesis by A5,FUNCT_2:3;
end;
