reserve y for set,
  x,a for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2 for PartFunc of REAL,REAL;

theorem Th31:
  Z c= dom (( #Z 2)*(f-exp_R)) & (for x st x in Z holds f.x=1)
implies ( #Z 2)*(f-exp_R) is_differentiable_on Z & for x st x in Z holds ((( #Z
  2)*(f-exp_R))`|Z).x =-2*exp_R.x*(1-exp_R.x)
proof
  assume that
A1: Z c= dom (( #Z 2)*(f-exp_R)) and
A2: for x st x in Z holds f.x=1;
  for y being object st y in Z holds y in dom (f-exp_R) by A1,FUNCT_1:11;
  then
A3: Z c= dom (f-exp_R) by TARSKI:def 3;
  then Z c= dom exp_R /\ dom f by VALUED_1:12;
  then
A4: Z c= dom f by XBOOLE_1:18;
A5: for x st x in Z holds f.x=0*x+1 by A2;
  then
A6: f is_differentiable_on Z by A4,FDIFF_1:23;
A7: for x st x in Z holds ( #Z 2)*(f-exp_R) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A8: f is_differentiable_in x by A6,FDIFF_1:9;
    exp_R is_differentiable_in x by SIN_COS:65;
    then f-exp_R is_differentiable_in x by A8,FDIFF_1:14;
    hence thesis by TAYLOR_1:3;
  end;
  then
A9: ( #Z 2)*(f-exp_R) is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((( #Z 2)*(f-exp_R))`|Z).x =-2*exp_R.x*(1-exp_R.x )
  proof
    let x;
A10: exp_R is_differentiable_in x by SIN_COS:65;
    assume
A11: x in Z;
    then
A12: (f-exp_R).x=f.x-exp_R.x by A3,VALUED_1:13
      .=1- exp_R.x by A2,A11;
A13: f is_differentiable_in x by A6,A11,FDIFF_1:9;
    then
A14: diff((f-exp_R),x)=diff(f,x)-diff(exp_R,x) by A10,FDIFF_1:14
      .=(f`|Z).x-diff(exp_R,x) by A6,A11,FDIFF_1:def 7
      .=(f`|Z).x-exp_R.x by SIN_COS:65
      .=0-exp_R.x by A4,A5,A11,FDIFF_1:23
      .=-exp_R.x;
A15: f-exp_R is_differentiable_in x by A13,A10,FDIFF_1:14;
    ((( #Z 2)*(f-exp_R))`|Z).x =diff((( #Z 2)*(f-exp_R)),x) by A9,A11,
FDIFF_1:def 7
      .=2* ((f-exp_R).x) #Z (2-1) * diff((f-exp_R),x) by A15,TAYLOR_1:3
      .=2*(1-exp_R.x)*(-exp_R.x) by A14,A12,PREPOWER:35
      .=-2*exp_R.x*(1-exp_R.x);
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:9;
end;
