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

theorem Th32:
  for f be Element of the carrier of Polynom-Ring INT.Ring holds
    ((exp_R1)(#)('F'(f)))|[.0,x0.] is continuous
    proof
      let f be Element of the carrier of Polynom-Ring INT.Ring;
      set f1 = (exp_R^), f2 = 'F'(f);
      dom f1 = REAL & dom f2 = REAL by Lm23,Lm23A; then
A3:   dom (f2|[.0,x0.]) = [.0,x0.] by RELAT_1:62;
A2:   dom (f1|[.0,x0.]) = [.0,x0.] by Lm23A, RELAT_1:62;
A4:   (f1(#)f2)|[.0,x0.] = f1|[.0,x0.] (#) f2|[.0,x0.] by RFUNCT_1:45;
      for r be Real st r in dom ((f1(#)f2)|[.0,x0.]) holds
      (f1(#)f2)|[.0,x0.] is_continuous_in r
      proof
        let r be Real;
        assume
A5:     r in dom ((f1(#)f2)|[.0,x0.]);
A6:     dom ((f1(#)f2)|[.0,x0.])
        = dom (f1|[.0,x0.] (#) f2|[.0,x0.]) by RFUNCT_1:45
        .= dom (f1|[.0,x0.]) /\ dom (f2|[.0,x0.]) by VALUED_1:def 4
        .= [.0,x0.] by A2,A3; then
A7:     f1|[.0,x0.] is_continuous_in r by A2,A5,Lm29,FCONT_1:def 2;
A8:     f2|[.0,x0.] is_continuous_in r by A5,A6,A3,Lm30,FCONT_1:def 2;
        r in dom (f1|[.0,x0.]) /\ dom (f2|[.0,x0.]) by A2,A3,A5,A6;
        hence thesis by A4,A7,A8,FCONT_1:7;
      end;
      hence thesis by FCONT_1:def 2;
    end;
