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 ((1/(1+log(number_e,a)))(#)((exp_R*f)(#)exp_R)) & (for x st x
  in Z holds f.x=x*log(number_e,a)) & a>0 & a<>1/number_e implies (1/(1+log(
  number_e,a)))(#)((exp_R*f)(#)exp_R) is_differentiable_on Z & for x st x in Z
  holds (((1/(1+log(number_e,a)))(#)((exp_R*f)(#)exp_R))`|Z).x =a #R x*exp_R.x
proof
  assume that
A1: Z c= dom ((1/(1+log(number_e,a)))(#)((exp_R*f)(#)exp_R)) and
A2: for x st x in Z holds f.x=x*log(number_e,a) and
A3: a>0 and
A4: a<>1/number_e;
A5: Z c= dom ((exp_R*f)(#)exp_R) by A1,VALUED_1:def 5;
  then Z c= dom (exp_R*f) /\ dom exp_R by VALUED_1:def 4;
  then
A6: Z c= dom (exp_R*f) by XBOOLE_1:18;
  then
A7: exp_R*f is_differentiable_on Z by A2,A3,Th11;
A8: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  then
A9: (exp_R*f)(#)exp_R is_differentiable_on Z by A5,A7,FDIFF_1:21;
A10: 1+log(number_e,a)<>0
  proof
A11: number_e*a>0*a by A3,TAYLOR_1:11,XREAL_1:68;
    assume
A12: 1+log(number_e,a)=0;
A13: number_e<>1 by TAYLOR_1:11;
    log(number_e,1)=0 by SIN_COS2:13,TAYLOR_1:13
      .=log(number_e,number_e)+log(number_e,a) by A12,A13,POWER:52,TAYLOR_1:11
      .=log(number_e,number_e*a) by A3,A13,POWER:53,TAYLOR_1:11;
    then number_e*a=(number_e) to_power log(number_e,1) by A13,A11,POWER:def 3
,TAYLOR_1:11
      .=1 by A13,POWER:def 3,TAYLOR_1:11;
    hence contradiction by A4,XCMPLX_1:73;
  end;
  for x st x in Z holds (((1/(1+log(number_e,a)))(#)((exp_R*f)(#)exp_R))
  `|Z).x =a #R x*exp_R.x
  proof
    let x;
    assume
A14: x in Z;
    then
A15: (exp_R*f).x=exp_R.(f.x) by A6,FUNCT_1:12
      .=exp_R.(x*log(number_e,a)) by A2,A14
      .=a #R x by A3,Th1;
    (((1/(1+log(number_e,a)))(#)((exp_R*f)(#)exp_R))`|Z).x =(1/(1+log(
    number_e,a)))*diff((exp_R*f)(#)exp_R,x) by A1,A9,A14,FDIFF_1:20
      .=(1/(1+log(number_e,a)))*(((exp_R*f)(#)exp_R)`|Z).x by A9,A14,
FDIFF_1:def 7
      .=(1/(1+log(number_e,a)))*(exp_R.x*diff(exp_R*f,x)+ (exp_R*f).x*diff(
    exp_R,x)) by A5,A7,A8,A14,FDIFF_1:21
      .=(1/(1+log(number_e,a)))*(exp_R.x*((exp_R*f)`|Z).x+ (exp_R*f).x*diff(
    exp_R,x)) by A7,A14,FDIFF_1:def 7
      .=(1/(1+log(number_e,a)))*(exp_R.x*((exp_R*f)`|Z).x+(exp_R*f).x*exp_R.
    x) by TAYLOR_1:16
      .=(1/(1+log(number_e,a)))*((((exp_R*f)`|Z).x+(exp_R*f).x)*exp_R.x)
      .=(1/(1+log(number_e,a)))*((a #R x*log(number_e,a)+(exp_R*f).x)*exp_R.
    x) by A2,A3,A6,A14,Th11
      .=(1/(1+log(number_e,a)))*(log(number_e,a)+1)*a #R x*exp_R.x by A15
      .=1*a #R x*exp_R.x by A10,XCMPLX_1:106
      .=a #R x*exp_R.x;
    hence thesis;
  end;
  hence thesis by A1,A9,FDIFF_1:20;
end;
