
theorem Th31:
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+(M,max+f) = integral+(CopyMeasure(T,M),max+g)
 & integral+(M,max-f) = integral+(CopyMeasure(T,M),max-g)
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: A = dom (max+f) & A = dom (max-f) by A3,MESFUNC2:def 2,def 3;
A6: max+f is A -measurable & max-f is A -measurable by A3,A4,MESFUNC2:25,26;
A7: max+g = (max+f)*T" & max-g = (max-f)*T" by A1,A2,Th27;
A8:for x being Element of X holds 0. <= (max+f).x by MESFUNC2:12;
    for x being Element of X holds 0. <= (max-f).x by MESFUNC2:13;
    hence thesis by A7,A8,Th29,A1,A5,A6,SUPINF_2:39;
end;
