reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  for f being Function,A be set st f is one-to-one & A c= bool dom f
  holds ("f)"A = (.:f).:A
proof
  let f be Function,A be set;
  assume f is one-to-one & A c= bool dom f;
  then ("f)"A c= (.:f).:A & (.:f).:A c= ("f)"A by Th31,Th32;
  hence thesis by XBOOLE_0:def 10;
end;
