
theorem Th28:
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 being Element of S
  st T is bijective & g = f*T" & A = dom f & f is A -measurable holds
   ex B being Element of CopyField(T,S) st
      B = T.:A & B = dom g & g is B -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, 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;

    dom (.:T) = bool X by FUNCT_2:def 1; then
    (.:T).A in (.:T).:S by FUNCT_1:def 6; then
    (.:T).A in CopyField(T,S) by A1,Def2; then
    reconsider B = T.:A as Element of CopyField(T,S) by A1,Th1;
    take B;
    dom g = (T")"(dom f) by A2,RELAT_1:147;
    hence thesis by A3,A1,A2,A4,Th20,FUNCT_1:84;
end;
