
theorem Th34:
for n be non zero Nat, X be non-empty n-element FinSequence,
    S be sigmaFieldFamily of X, m be sigmaMeasureFamily of S,
    f be PartFunc of CarProduct X,ExtREAL,
    g be PartFunc of product X,ExtREAL,
    A be Element of Prod_Field S, B be Element of XProd_Field S
 st B = (CarProd X).:A & g = f*(CarProd X)" holds
    f is A -measurable iff g is B -measurable
proof
    let n be non zero Nat, X be non-empty n-element FinSequence,
    S be sigmaFieldFamily of X, m be sigmaMeasureFamily of S,
    f be PartFunc of CarProduct X,ExtREAL,
    g be PartFunc of product X,ExtREAL,
    A be Element of Prod_Field S, B be Element of XProd_Field S;
    assume
A1: B = (CarProd X).:A & g = f*(CarProd X)";
    CarProd X is bijective by Th12;
    hence thesis by Th20,A1;
end;
