
theorem
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, B being Element of CopyField(T,S)
  st T is bijective & g = f*T" & B = dom g & g is B -measurable holds
   ex A being Element of S st B = T.:A & A = dom f & f is A -measurable
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, B being Element of CopyField(T,S);
    assume that
A1: T is bijective and
A2: g = f*T" and
A3: B = dom g and
A4: g is B -measurable;

    consider H be Function of Y,X such that
A5: H is bijective & H =T" & H" =T & .:H = (.:T)"
  & (.:H).:CopyField(T,S) = S & CopyField(H,CopyField(T,S)) = S by A1,Th17;

    T"B in S by Th19,A1; then
    reconsider A = H.:B as Element of S by A5,FUNCT_1:85;
    take A;
    dom f in bool X; then
A6: dom f in dom (.:T) by FUNCT_2:def 1;

    .:T is bijective by A1,Th1; then
A7: rng (.:T) = bool Y by FUNCT_2:def 3;

    H.:(dom g) = H.:((.:T).(dom f)) by A1,A2,Th15; then
A8: H.:(dom g) = (.:H).((.:T).(dom f)) by A5,Th1;

    T.:A = T.:((.:H).B) by A5,Th1; then
    T.:A = (.:T).((.:H).B) by A1,Th1; then
    T.:A = B by A5,A7,FUNCT_1:35;
    hence thesis by A8,A1,A2,A4,A3,A6,A5,Th20,FUNCT_1:34;
end;
