
theorem Th26:
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
  st T is bijective & g = f*T" & f is_simple_func_in S & f is nonnegative
  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;
    assume that
A1: T is bijective and
A2: g = f*T" and
A3: f is_simple_func_in S and
A4: f is nonnegative;

    set L = CopyMeasure(T,M);

A5: dom T = X by FUNCT_2:def 1;
A6: dom f c= X;
    g*T = f*(T"*T) by A2,RELAT_1:36; then
    g*T = f* (id dom T) by A1,FUNCT_1:39; then
A7: g*T = f by RELAT_1:51,A5,A6;

    per cases;
    suppose
A8:  dom f = {};
     now assume dom g <> {}; then
      ex x be object st x in dom g by XBOOLE_0:def 1;
      hence contradiction by A8,A2,FUNCT_1:11;
     end; then
     integral'(L,g) = 0. by MESFUNC5:def 14;
     hence integral'(L,g) = integral'(M,f) by A8,MESFUNC5:def 14;
    end;
    suppose
A9:  dom f <> {}; then
     ex x be object st x in dom f by XBOOLE_0:def 1; then
     dom g <> {} by A7,FUNCT_1:11; then
     integral'(L,g) = integral(L,g) by MESFUNC5:def 14; then
     integral'(L,g) = integral(M,f) by A1,A2,A3,A4,Th25;
     hence integral'(L,g) = integral'(M,f) by MESFUNC5:def 14,A9;
    end;
end;
