
theorem Th22:
for X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
 f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL
  st T is bijective & g = f*T" holds
   f is_simple_func_in S iff g is_simple_func_in CopyField(T,S)
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
    f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL;
    assume
A1: T is bijective & g = f*T";
    hence f is_simple_func_in S implies g is_simple_func_in CopyField(T,S)
      by Lm1;
    assume
A2: g is_simple_func_in CopyField(T,S);

    consider H be Function of Y,X such that
A3: H is bijective & H = T" & H" = T & .:H = (.:T) "
  & (.:H).:CopyField(T,S) = S & CopyField(H,CopyField(T,S)) = S by Th17,A1;

A4: dom T = X by FUNCT_2:def 1;
A5: dom f c= X;
    g*T = f*(T"*T) by A1,RELAT_1:36; then
    g*T = f* (id dom T) by FUNCT_1:39,A1; then
    g*T = f by RELAT_1:51,A4,A5;
    hence thesis by A2,A3,Lm1;
end;
