
theorem
for n be non zero Nat, f be PartFunc of CarProduct(Seg n --> REAL),ExtREAL,
  g be PartFunc of REAL n,ExtREAL, A be Element of Prod_Field(L-Field n)
 st g = f*(CarProd(Seg n --> REAL))" & A = dom f & f is A -measurable holds
   Integral(XL-Meas n,g) = Integral(Prod_Measure(L-Meas n),f)
proof
    let n be non zero Nat,
    f be PartFunc of CarProduct(Seg n --> REAL),ExtREAL,
    g be PartFunc of REAL n, ExtREAL, A be Element of Prod_Field(L-Field n);
    assume that
A1: g = f*(CarProd(Seg n --> REAL))" and
A2: A = dom f and
A3: f is A -measurable;
    product(Seg n --> REAL) = REAL n by SRINGS_5:8;
    hence thesis by A1,A2,A3,Th12,Th32;
end;
