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 Th23:
  Z c= dom ((( #Z 2)*exp_R)+f) & (for x st x in Z holds f.x=1)
  implies (( #Z 2)*exp_R)+f is_differentiable_on Z & for x st x in Z holds ((((
  #Z 2)*exp_R)+f)`|Z).x =2*exp_R(2*x)
proof
  assume that
A1: Z c= dom ((( #Z 2)*exp_R)+f) and
A2: for x st x in Z holds f.x=1;
A3: Z c= dom (( #Z 2)*exp_R) /\ dom f by A1,VALUED_1:def 1;
  then
A4: Z c= dom f by XBOOLE_1:18;
A5: now
    let x;
    assume x in Z;
    exp_R is_differentiable_in x by SIN_COS:65;
    hence ( #Z 2)*exp_R is_differentiable_in x by TAYLOR_1:3;
  end;
A6: for x st x in Z holds f.x=0*x+1 by A2;
  then
A7: f is_differentiable_on Z by A4,FDIFF_1:23;
  Z c= dom (( #Z 2)*exp_R) by A3,XBOOLE_1:18;
  then
A8: ( #Z 2)*exp_R is_differentiable_on Z by A5,FDIFF_1:9;
  for x st x in Z holds (((( #Z 2)*exp_R)+f)`|Z).x =2*exp_R(2*x)
  proof
    let x;
A9: exp_R is_differentiable_in x by SIN_COS:65;
    assume
A10: x in Z;
    then
    (((( #Z 2)*exp_R)+f)`|Z).x =diff(( #Z 2)*exp_R,x)+diff(f,x) by A1,A7,A8,
FDIFF_1:18
      .=(2*( (exp_R.x) #Z (2-1)) * diff(exp_R,x))+diff(f,x) by A9,TAYLOR_1:3
      .=2*( (exp_R.x) #Z (2-1)) * exp_R.x+diff(f,x) by SIN_COS:65
      .=2* exp_R.x * exp_R.x+diff(f,x) by PREPOWER:35
      .=2*( exp_R.x * exp_R.x)+diff(f,x)
      .=2*( exp_R( x )* exp_R.x)+diff(f,x) by SIN_COS:def 23
      .=2*( exp_R( x )* exp_R(x))+diff(f,x) by SIN_COS:def 23
      .=2*(exp_R(x+x))+diff(f,x) by SIN_COS:50
      .=2* exp_R(2*x)+(f`|Z).x by A7,A10,FDIFF_1:def 7
      .= 2* exp_R(2*x)+0 by A4,A6,A10,FDIFF_1:23
      .=2* exp_R(2*x);
    hence thesis;
  end;
  hence thesis by A1,A7,A8,FDIFF_1:18;
end;
