
theorem
for f be PartFunc of [:REAL,REAL:],ExtREAL, g be PartFunc of REAL 2,ExtREAL,
 A be Element of sigma measurable_rectangles(L-Field,L-Field),
 B be Element of XL-Field 2
 st g = f*(CarProd(Seg 2 --> REAL))" & B = (CarProd(Seg 2 --> REAL)).:A
 holds f is A-measurable iff g is B-measurable
proof
    let f be PartFunc of [:REAL,REAL:],ExtREAL,
    g be PartFunc of REAL 2,ExtREAL,
    A be Element of sigma measurable_rectangles(L-Field,L-Field),
    B be Element of XL-Field 2;
    assume that
A1: g = f*(CarProd(Seg 2 --> REAL))" and
A2: B = (CarProd(Seg 2 --> REAL)).:A;

    Prod_Field(L-Field (1+1))
     = sigma measurable_rectangles(Prod_Field(L-Field 1),L-Field) by Th44; then
    reconsider A1 = A as Element of Prod_Field(L-Field 2) by Th37,Th41;
    reconsider f1 = f as PartFunc of CarProduct(Seg 2 --> REAL),ExtREAL
      by Th37;
    f1 is A1-measurable iff g is B-measurable by A1,A2,Th49;
    hence f is A-measurable iff g is B-measurable by Th50;
end;
