reserve x,y,z,X,Y for set;

theorem Th3:
  for X,Y for f being Function holds f.:(Y \ f"X) = f.:Y \ X
proof
  let X,Y;
  let f be Function;
  now
    let x be object;
    thus x in f.:(Y \ f"X) implies x in f.:Y \ X
    proof
      assume x in f.:(Y \ f"X);
      then consider z being object such that
A1:   z in dom f and
A2:   z in Y \ f"X and
A3:   f.z = x by FUNCT_1:def 6;
      not z in f"X by A2,XBOOLE_0:def 5;
      then
A4:   not x in X by A1,A3,FUNCT_1:def 7;
      f.z in f.:Y by A1,A2,FUNCT_1:def 6;
      hence thesis by A3,A4,XBOOLE_0:def 5;
    end;
    assume
A5: x in f.:Y \ X;
    then consider z being object such that
A6: z in dom f and
A7: z in Y and
A8: f.z = x by FUNCT_1:def 6;
    not x in X by A5,XBOOLE_0:def 5;
    then not z in f"X by A8,FUNCT_1:def 7;
    then z in Y \ f"X by A7,XBOOLE_0:def 5;
    hence x in f.:(Y \ f"X) by A6,A8,FUNCT_1:def 6;
  end;
  hence thesis by TARSKI:2;
end;
