
theorem Th52:
for f1 be PartFunc of [:[:REAL,REAL:],REAL:],ExtREAL,
    f2 be PartFunc of CarProduct(Seg 3 --> REAL),ExtREAL,
    A1 be Element of sigma measurable_rectangles(
       sigma measurable_rectangles(L-Field,L-Field),L-Field),
    A2 be Element of Prod_Field(L-Field 3) st f1 = f2 & A1 = A2 holds
  f1 is A1-measurable iff f2 is A2-measurable
proof
    let f1 be PartFunc of [:[:REAL,REAL:],REAL:],ExtREAL,
    f2 be PartFunc of CarProduct(Seg 3 --> REAL),ExtREAL,
    A1 be Element of sigma measurable_rectangles(
       sigma measurable_rectangles(L-Field,L-Field),L-Field),
    A2 be Element of Prod_Field(L-Field 3);
    assume that
A1: f1 = f2 and
A2: A1 = A2;
    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;
    hence thesis by A1,A2,Th37,Th41;
end;
