
theorem Th20:
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,
 A be Element of S, B be Element of CopyField(T,S)
  st T is bijective & B = T.:A & g = f*T" holds
   f is A -measurable iff g is B -measurable
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,
    A be Element of S, B be Element of CopyField(T,S);
    assume that
A1: T is bijective and
A2: B = T.:A and
A3: g = f*T";

A4: dom T = X by FUNCT_2:def 1;
A5: dom f c= X;
    g*T = f*(T"*T) by A3,RELAT_1:36; then
    g*T = f* (id dom T) by A1,FUNCT_1:39; then
A6: g*T = f by A4,A5,RELAT_1:51;

    consider H be Function of Y,X such that
A7: H is bijective & H = T" & H" = T & .:H = (.:T) "
  & (.:H).: CopyField(T,S) = S & CopyField(H,CopyField(T,S)) = S by A1,Th17;
    H.:B = T"(T.:A) by A2,A7,FUNCT_1:85; then
A8: H.:B = A by A1,A4,FUNCT_1:94;

    hereby assume
A9:  f is A -measurable;
     now let r be Real;
A10:  A /\ less_dom(f,r) in S by A9;

      (.:T).(A /\ less_dom(f,r)) = T.:(A /\ less_dom(f,r)) by A1,Th1; then
      (.:T).(A /\ less_dom(f,r)) = (T.:A) /\ T.:(less_dom(f,r))
        by A1,FUNCT_1:62; then
A11:  (.:T).(A /\ less_dom(f,r)) = B /\ less_dom(g,r) by A2,A1,A3,Th16;

      A /\ less_dom(f,r) in bool X; then
      A /\ less_dom(f,r) in dom (.:T) by FUNCT_2:def 1; then
      B /\ less_dom(g,r) in (.:T).:S  by A10,A11,FUNCT_1:def 6;
      hence B /\ less_dom(g,r) in CopyField(T,S) by A1,Def2;
     end;
     hence g is B -measurable;
    end;

    assume
A12:g is B -measurable;

    now let r be Real;
A13: B /\ less_dom(g,r) in CopyField(T,S) by A12;

     (.:H).(B /\ less_dom(g,r)) = H.:(B /\ less_dom(g,r)) by A7,Th1; then
     (.:H).(B /\ less_dom(g,r)) = (H.:B) /\ H.:(less_dom(g,r))
       by A7,FUNCT_1:62; then
A14: (.:H).(B /\ less_dom(g,r)) = A /\ less_dom(f,r) by A8,A6,A7,Th16;

     B /\ less_dom(g,r) in bool Y; then
     B /\ less_dom(g,r) in dom (.:H) by FUNCT_2:def 1;
     hence A /\ less_dom(f,r) in S by A7,A13,A14,FUNCT_1:def 6;
    end;
    hence f is A -measurable;
end;
