
theorem
for f be PartFunc of [:[:REAL,REAL:],REAL:],ExtREAL,
 g be PartFunc of REAL 3,ExtREAL,
 A be Element of sigma measurable_rectangles(
       sigma measurable_rectangles(L-Field,L-Field),L-Field),
 B be Element of XL-Field 3
 st g = f*(CarProd(Seg 3 --> REAL))" & B = (CarProd(Seg 3 --> REAL)).:A
 holds f is A-measurable iff g is B-measurable
proof
    let f be PartFunc of [:[:REAL,REAL:],REAL:],ExtREAL,
    g be PartFunc of REAL 3,ExtREAL,
    A be Element of sigma measurable_rectangles(
       sigma measurable_rectangles(L-Field,L-Field),L-Field),
    B be Element of XL-Field 3;
    assume that
A1: g = f*(CarProd(Seg 3 --> REAL))" and
A2: B = (CarProd(Seg 3 --> REAL)).:A;

    Prod_Field(L-Field (2+1))
     = sigma measurable_rectangles(Prod_Field(L-Field 2),L-Field)
  & 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 3) by Th37,Th41;

    set Z = Seg 3 --> REAL;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3; then
A3: Z.1 = REAL & Z.2 = REAL & Z.3 = REAL by FUNCOP_1:7;
    (ProdFinSeq Z).1 = Z.1
  & (ProdFinSeq Z).2 = [: (ProdFinSeq Z).1,Z.(1+1) :]
  & (ProdFinSeq Z).3 = [: (ProdFinSeq Z).2,Z.(2+1) :] by MEASUR13:def 3; then
    reconsider f1 = f as PartFunc of CarProduct(Seg 3 --> REAL),ExtREAL by A3;
    f1 is A1-measurable iff g is B-measurable by A1,A2,Th49;
    hence f is A-measurable iff g is B-measurable by Th52;
end;
