
theorem Th33:
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" holds
  f is_integrable_on M iff g is_integrable_on CopyMeasure(T,M)
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
A1: T is bijective & g = f*T";

    hereby assume
A2:  f is_integrable_on M; then
A3:  integral+(M,max+f) < +infty
   & integral+(M,max-f) < +infty by MESFUNC5:def 17;
     consider A be Element of S such that
A4:  A = dom f & f is A-measurable by A2,MESFUNC5:def 17;
     consider B be Element of CopyField(T,S) such that
A5:  B = T.:A & B = dom g & g is B -measurable by A1,A4,Th28;
     integral+(M,max+f) = integral+(CopyMeasure(T,M),max+g)
   & integral+(M,max-f) = integral+(CopyMeasure(T,M),max-g)
        by A4,Th31,A1;
     hence g is_integrable_on CopyMeasure(T,M) by A3,A5,MESFUNC5:def 17;
    end;
    assume
A6: g is_integrable_on CopyMeasure(T,M); then
A7: integral+(CopyMeasure(T,M),max+g) < +infty
  & integral+(CopyMeasure(T,M),max-g) < +infty by MESFUNC5:def 17;

    consider B being Element of CopyField(T,S) such that
A8: B = dom g & g is B -measurable by A6,MESFUNC5:def 17;

    consider H be Function of Y,X such that
A9: H is bijective & H=T" & H"=T & .:H = (.:T)"
  & (.:H).:CopyField(T,S) = S & CopyField(H,CopyField(T,S)) = S by Th17,A1;

    dom (.:H) = bool Y by FUNCT_2:def 1; then
    (.:H).B in S by A9,FUNCT_1:def 6; then
    reconsider A = H.:B as Element of S by A9,Th1;

    .:T is bijective by A1,Th1; then
A10: rng (.:T) = bool Y by FUNCT_2:def 3;

    T.:A = (.:T).A by A1,Th1; then
    T.:A = (.:T).((.:H).B) by A9,Th1; then
    T.:A = B by A10,A9,FUNCT_1:35; then
A11: f is A -measurable by Th20,A1,A8;
    dom f in bool X; then
A12: dom f in dom (.:T) by FUNCT_2:def 1;

    B = (T")"(dom f) by A1,A8,RELAT_1:147; then
    B = T.:(dom f) by A1,FUNCT_1:84; then
    B = (.:T).(dom f) by A1,Th1; then
    (.:H).B = dom f by A9,A12,FUNCT_1:34; then
A13:A = dom f by A9,Th1; then
    integral+(M,max+f) = integral+(CopyMeasure(T,M),max+g)
  & integral+(M,max-f) = integral+(CopyMeasure(T,M),max-g) by Th31,A1,A11;
    hence f is_integrable_on M by MESFUNC5:def 17,A7,A11,A13;
end;
