
theorem Th32:
for X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
 M be sigma_Measure of S, f be PartFunc of X,ExtREAL,
 g be PartFunc of Y,ExtREAL, A be Element of S
  st T is bijective & g = f*T" & A = dom f & f is A -measurable
 holds Integral(CopyMeasure(T,M),g) = Integral(M,f)
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
    M be sigma_Measure of S, f be PartFunc of X,ExtREAL,
    g be PartFunc of Y,ExtREAL, A be Element of S;
    assume that
A1: T is bijective and
A2: g = f*T" and
A3: A = dom f and
A4: f is A -measurable;

A5: integral+(M,max+f) = integral+(CopyMeasure(T,M),max+g)
  & integral+(M,max-f) = integral+(CopyMeasure(T,M),max-g) by A1,A2,A3,A4,Th31;

    Integral(M,f) = integral+(M,max+f) - integral+(M,max-f)
      by MESFUNC5:def 16;
    hence thesis by A5,MESFUNC5:def 16;
end;
