
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, G2 be PartFunc of REAL,REAL
 st [:I,J:] = dom f & f is_continuous_on [:I,J:] & f = g
  & G2 = Integral2(L-Meas,R_EAL g)|I holds
   Integral(Prod_Measure(L-Meas,L-Meas),R_EAL g) = integral(G2,I)
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, G2 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:  G2 = Integral2(L-Meas,R_EAL g)|I;

    set Rg =R_EAL g;
    set NI = REAL \ I;
    set RG2 = Integral2(L-Meas,Rg);
    set F0 = RG2|I, F1 = RG2|NI;

A5:dom RG2 = REAL by FUNCT_2:def 1; then
A6:dom G2 = I by A4;

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

    G2 is continuous by A1,A2,A3,A4,Th53; then
    G2|I is bounded & G2 is_integrable_on I by A4,A5,INTEGRA5:10,11; then
A8:Integral(L-Meas,G2|I) = integral(G2,I) by A6,A7,MESFUN14:49;

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

A10:Integral(Prod_Measure(L-Meas,L-Meas),g)
     = Integral(L-Meas,Integral2(L-Meas,R_EAL g)) by A2,A1,A3,Lm4;

    I \/ NI = REAL by XBOOLE_1:45; then
A11:RG2|(I \/ NI) = RG2;

    RG2 is_integrable_on L-Meas by A2,A3,A1,Lm4; then
A12:Integral(L-Meas,RG2) = Integral (L-Meas,F0) + Integral (L-Meas,F1)
      by A7,A9,A11,XBOOLE_1:85,MESFUNC5:98;

    for x being Element of REAL st x in dom F1 holds F1.x = 0
    proof
     let x be Element of REAL;
     assume
A13: x in dom F1; then
     not x in I by XBOOLE_0:def 5; then
A14: dom ProjPMap1(Rg,x) = {} by A1,A3,Th27;
     (Integral2(L-Meas,Rg)).x = Integral(L-Meas,(ProjPMap1(Rg,x)))
       by MESFUN12:def 8; then
     (Integral2 (L-Meas,Rg)).x = 0 by A14,Th1;
     hence thesis by A13,FUNCT_1:49;
    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;
