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
  Z c= dom ((2/3)(#)(( #R (3/2))*(f+exp_R))) & (for x st x in Z holds f.
x=1) implies (2/3)(#)(( #R (3/2))*(f+exp_R)) is_differentiable_on Z & for x st
x in Z holds (((2/3)(#)(( #R (3/2))*(f+exp_R)))`|Z).x =exp_R.x*(1+exp_R.x) #R (
  1/2)
proof
  assume that
A1: Z c= dom ((2/3)(#)(( #R (3/2))*(f+exp_R))) and
A2: for x st x in Z holds f.x=1;
A3: for x st x in Z holds f.x=0*x+1 by A2;
A4: Z c= dom (( #R (3/2))*(f+exp_R)) by A1,VALUED_1:def 5;
  then for y being object st y in Z holds y in dom (f+exp_R) by FUNCT_1:11;
  then
A5: Z c= dom (f+exp_R) by TARSKI:def 3;
  then Z c= dom exp_R /\ dom f by VALUED_1:def 1;
  then
A6: Z c= dom f by XBOOLE_1:18;
  then
A7: f is_differentiable_on Z by A3,FDIFF_1:23;
A8: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  then
A9: f+exp_R is_differentiable_on Z by A5,A7,FDIFF_1:18;
A10: for x st x in Z holds (f+exp_R).x>0
  proof
    let x;
    assume
A11: x in Z;
    then (f+exp_R).x=f.x+exp_R.x by A5,VALUED_1:def 1
      .=1+exp_R.x by A2,A11;
    hence thesis by SIN_COS:54,XREAL_1:34;
  end;
  now
    let x;
    assume x in Z;
    then f+exp_R is_differentiable_in x & (f+exp_R).x>0 by A9,A10,FDIFF_1:9;
    hence ( #R (3/2))*(f+exp_R) is_differentiable_in x by TAYLOR_1:22;
  end;
  then
A12: ( #R (3/2))*(f+exp_R) is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds (((2/3)(#)(( #R (3/2))*(f+exp_R)))`|Z).x =exp_R.x
  *(1+exp_R.x) #R (1/2)
  proof
    let x;
    assume
A13: x in Z;
    then
A14: ((f+exp_R)`|Z).x = diff(f,x)+diff(exp_R,x) by A5,A7,A8,FDIFF_1:18
      .=diff(f,x)+exp_R.x by SIN_COS:65
      .=(f`|Z).x+exp_R.x by A7,A13,FDIFF_1:def 7
      .=0+exp_R.x by A6,A3,A13,FDIFF_1:23
      .=exp_R.x;
A15: f+exp_R is_differentiable_in x & (f+exp_R).x>0 by A9,A10,A13,FDIFF_1:9;
A16: (f+exp_R).x=f.x+exp_R.x by A5,A13,VALUED_1:def 1
      .=1+exp_R.x by A2,A13;
    (((2/3)(#)(( #R (3/2))*(f+exp_R)))`|Z).x =(2/3)*diff((( #R (3/2))*(f+
    exp_R)),x) by A1,A12,A13,FDIFF_1:20
      .=(2/3)*((3/2)*(((f+exp_R).x) #R (3/2-1)) * diff((f+exp_R),x)) by A15,
TAYLOR_1:22
      .=(2/3)*((3/2)*(((f+exp_R).x) #R (3/2-1)) *((f+exp_R)`|Z).x ) by A9,A13,
FDIFF_1:def 7
      .=exp_R.x*(1+exp_R.x) #R (1/2) by A16,A14;
    hence thesis;
  end;
  hence thesis by A1,A12,FDIFF_1:20;
end;
