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

theorem
  for f be Element of the carrier of Polynom-Ring INT.Ring,
    x0 be positive Real st len f > 0 holds
    ex s be Real st 0 < s & s < 1 &
    ('F'(f)).x0 - (exp_R.x0)*(('F'(f)).0)
    = -x0*(exp_R.(x0*(1-s)))*(Eval(~(^f))).(s*x0)
    proof
      let f be Element of the carrier of Polynom-Ring INT.Ring,
          x0 be positive Real;
      assume
A1:   len f > 0;
      set PHI = (exp_R1)(#)('F'(f));
      consider s be Real such that
A2:   0 < s & s < 1 and
A3:   (PHI).x0 = (PHI).0 + x0*diff(PHI,s*x0) by Lm33;
A4:   (PHI).x0 = (exp_R1.x0)*('F'(f)).x0 by VALUED_1:5;
A5:   (PHI).0 = (exp_R1.0)*('F'(f)).0 by VALUED_1:5
      .= (exp_R.0)"*('F'(f)).0 by RFUNCT_1:59
      .= 1"*('F'(f)).0 by SIN_COS:51,def 23
      .= ('F'(f)).0;
reconsider PHi = PHI as differentiable Function of REAL,REAL;
A6:   diff(PHI,s*x0) = (PHi`|).(s*x0) by POLYDIFF:10
      .=( -(exp_R1(#)Eval(~(^f)))).(s*x0) by A1,Lm36
      .= -((exp_R1(#)Eval(~(^f))).(s*x0)) by VALUED_1:8
      .= -(exp_R1.(s*x0)*(Eval(~(^f))).(s*x0)) by VALUED_1:5;
A7:   x0 in dom (exp_R^) by Lm23A,XREAL_0:def 1;
A8:   exp_R.x0 <> 0 by SIN_COS:54;
A9:   x0*s in dom (exp_R^) by Lm23A,XREAL_0:def 1;
      reconsider x1 = x0*s as Real;
      reconsider r = exp_R.x0 as Real;
A10:  exp_R1.(s*x0)
      = 1/exp_R.x1 by A9,RFUNCT_1:59 .= 1/exp_R(x1) by SIN_COS:def 23
      .= exp_R(-x1) by TAYLOR_1:4 .= exp_R.(-x1) by SIN_COS:def 23;
      ('F'(f)).x0 - (exp_R.x0)*(('F'(f)).0)
      = (1*('F'(f)).x0) - (exp_R.x0)*(('F'(f)).0)
      .= (r*r")*('F'(f)).x0 - (exp_R.x0)*(('F'(f)).0) by A8,XCMPLX_0:def 7
      .= ((exp_R.x0)*(exp_R1.x0))*('F'(f)).x0
         - (exp_R.x0)*(('F'(f)).0) by A7,RFUNCT_1:59
      .= (exp_R.x0)*(PHI.x0 - PHI.0) by A5,A4
      .= -x0*(exp_R.x0 * exp_R.(-(s*x0)))*(Eval(~(^f))).(s*x0) by A10,A6,A3
      .= -x0*(exp_R.(x0+(-(s*x0))))*(Eval(~(^f))).(s*x0) by SIN_COS2:12
      .= -x0*(exp_R.(x0*(1-s)))*(Eval(~(^f))).(s*x0);
      hence thesis by A2;
    end;
