
theorem
for I,J be non empty closed_interval Subset of REAL,
 f being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g being PartFunc of [:REAL,REAL:],REAL, G1 be PartFunc of REAL,REAL
 st [:I,J:] = dom f & f is_continuous_on [:I,J:] & f = g
  & G1 = Integral1(L-Meas,R_EAL g)|J holds
   Integral(Prod_Measure(L-Meas,L-Meas),R_EAL g) = integral(G1,J)
proof
    let I,J be non empty closed_interval Subset of REAL,
    f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, G1 be PartFunc of REAL,REAL;
    assume that
A1: [:I,J:] = dom f and
A2: f is_continuous_on [:I,J:] and
A3: f = g and
A4: G1 = Integral1(L-Meas,R_EAL g)|J;

    set Rg =R_EAL g;
    set NJ = REAL \ J;
    set RG1 = Integral1(L-Meas,Rg);
    set F0 = RG1|J, F1 = RG1|NJ;

A5:dom RG1 = REAL by FUNCT_2:def 1; then
A6:dom(RG1|J) = J;

A7:J is Element of L-Field by MEASUR10:5,MEASUR12:75;

    G1|J is bounded & G1 is_integrable_on J
      by A6,A1,A2,A3,A4,Th56,INTEGRA5:10,11; then
A8:Integral(L-Meas,G1|J) = integral(G1,J) by A4,A7,A6,MESFUN14:49;

    REAL in L-Field by PROB_1:5; then
A9: NJ is Element of L-Field by A7,PROB_1:6;

A10:Integral(Prod_Measure(L-Meas,L-Meas),g)
     = Integral(L-Meas,Integral1(L-Meas,R_EAL g)) by A2,A1,A3,Lm5;

    J \/ NJ = REAL by XBOOLE_1:45; then
A11:RG1|(J \/ NJ) = RG1;

    RG1 is_integrable_on L-Meas by A2,A1,A3,Lm5; then
A12:Integral(L-Meas,RG1) = Integral(L-Meas,F0) + Integral(L-Meas,F1)
      by A7,A9,A11,XBOOLE_1:85,MESFUNC5:98;

    for y being Element of REAL st y in dom F1 holds F1.y = 0
    proof
     let y be Element of REAL;
     assume
A13: y in dom F1; then
     not y in J by XBOOLE_0:def 5; then
A14: dom ProjPMap2(Rg,y) = {} by A1,A3,Th28;
     (Integral1(L-Meas,Rg)).y = Integral(L-Meas,(ProjPMap2(Rg,y)))
       by MESFUN12:def 7; then
     (Integral1 (L-Meas,Rg)).y = 0 by A14,Th1;
     hence thesis by FUNCT_1:49,A13;
    end; then
    Integral(L-Meas,F1) = 0 by A9,A5,MESFUN12:57; then
    Integral(Prod_Measure(L-Meas,L-Meas),Rg) = Integral(L-Meas,F0)
      by A10,A12,XXREAL_3:4;
    hence thesis by A4,A8,MESFUNC5:def 7;
end;
