
theorem Th41:
for I,J,K be non empty closed_interval Subset of REAL,
  z be Element of REAL,
  f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
  g being 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.|).z < +infty
proof
    let I,J,K be non empty closed_interval Subset of REAL,
    y be Element 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;

    dom(Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)) = REAL
      by FUNCT_2:def 1; then
A4: dom(Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)|K) = K;

    reconsider G = Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)|K
      as PartFunc of REAL,REAL by A1,A2,A3,Th35;

    per cases;
    suppose
A5:  y in K; then
     Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).y = G.y
       by FUNCT_1:49; then
     Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).y in rng G
       by A4,A5,FUNCT_1:3;
     hence Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).y < +infty
       by XXREAL_0:9;
    end;
    suppose not y in K; then
     dom ProjPMap2(|.R_EAL g.|,y) = {} by A1,A3,MESFUN16:28; then
     Integral(Prod_Measure(L-Meas,L-Meas),ProjPMap2(|.R_EAL g.|,y)) = 0
       by MESFUN16:1; then
     Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).y = 0
       by MESFUN12:def 7;
     hence Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).y < +infty
       by XREAL_0:def 1,XXREAL_0:9;
    end;
end;
