
theorem Th4:
  for A being non empty set, y being set, f being Function holds (f
  +*(A --> y)).:A = {y}
proof
  let A be non empty set, y be set, f be Function;
  now
    let u be object;
    thus u in (f+*(A --> y)).:A implies u = y
    proof
      assume u in (f+*(A --> y)).:A;
      then consider z being object such that
      z in dom(f+*(A --> y)) and
A1:   z in A and
A2:   u = (f+*(A --> y)).z by FUNCT_1:def 6;
      z in dom(A --> y) by A1,FUNCOP_1:13;
      then u = (A --> y).z by A2,FUNCT_4:13;
      hence thesis by A1,FUNCOP_1:7;
    end;
    consider x being object such that
A3: x in A by XBOOLE_0:def 1;
A4: x in dom(A --> y) by A3,FUNCOP_1:13;
    then
A5: x in dom(f+*(A --> y)) by FUNCT_4:12;
    (A --> y).x = y by A3,FUNCOP_1:7;
    then y = (f+*(A --> y)).x by A4,FUNCT_4:13;
    hence u = y implies u in (f+*(A --> y)).:A by A3,A5,FUNCT_1:def 6;
  end;
  hence thesis by TARSKI:def 1;
end;
