
theorem Th18:
  for X,Y be non empty set,
      S be SigmaField of X, T be Function of X,Y,
      A be Subset of X st T is bijective holds
    A in S iff T.:A in CopyField(T,S)
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
        A be Subset of X;
    assume
A1: T is bijective;

A2: dom (.:T) = bool X by FUNCT_2:def 1;

    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 A1,Th17;

A4: dom (.:H) = bool Y by FUNCT_2:def 1;

    hereby assume A in S; then
     (.:T).A in (.:T).:S by A2,FUNCT_1:def 6; then
     (.:T).A in CopyField(T,S) by A1,Def2;
     hence T.:A in CopyField(T,S) by A1,Th1;
    end;
    assume T.:A in CopyField(T,S); then
    (.:T).A in CopyField(T,S) by A1,Th1; then
    (.:H).((.:T).A) in (.:H).:CopyField(T,S) by A4,FUNCT_1:def 6;
    hence A in S by A2,A3,FUNCT_1:34;
end;
