 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th33:
  for f be Element of the carrier of Polynom-Ring INT.Ring holds
    (exp_R1)(#)('F'(f)) is_differentiable_on ].0,x0.[
     proof
       let f be Element of the carrier of Polynom-Ring INT.Ring;
       set f1 = exp_R^, f2 = 'F'(f);
       set Z = ].0,x0.[;
A1:    dom f1 = REAL & dom f2 = REAL by Lm23A,Lm23;
A2:    f1 is_differentiable_on Z by Lm31;
A3:    f2 is_differentiable_on Z by Lm32;
       dom (f1(#)f2) = dom f1 /\ dom f2 by VALUED_1:def 4 .= REAL by A1;
       hence thesis by A2,A3,FDIFF_1:21;
     end;
