theorem
  (for x holds x in X iff x in Y & x in Z) implies X = Y (/\) Z
proof
  assume
A1: for x holds x in X iff x in Y & x in Z;
  now
    let x;
    hereby
      assume x in X;
      then x in Y & x in Z by A1;
      hence x in Y (/\) Z by Th8;
    end;
    assume x in Y (/\) Z;
    then x in Y & x in Z by Th8;
    hence x in X by A1;
  end;
  hence thesis by Th135;
end;
