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/(log(number_e,a)-1))(#)((exp_R*f)/exp_R)) & (for x st x
in Z holds f.x=x*log(number_e,a)) & a>0 & a<>number_e implies (1/(log(number_e,
a)-1))(#)((exp_R*f)/exp_R) is_differentiable_on Z & for x st x in Z holds (((1/
  (log(number_e,a)-1))(#)((exp_R*f)/exp_R))`|Z).x =a #R x/exp_R.x
proof
  assume that
A1: Z c= dom ((1/(log(number_e,a)-1))(#)((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<>number_e;
  Z c= dom ((exp_R*f)/exp_R) by A1,VALUED_1:def 5;
  then Z c= dom (exp_R*f) /\ (dom exp_R \ (exp_R)"{0}) by RFUNCT_1:def 1;
  then
A5: Z c= dom (exp_R*f) by XBOOLE_1:18;
  then
A6: exp_R*f is_differentiable_on Z by A2,A3,Th11;
  exp_R is_differentiable_on Z & for x st x in Z holds exp_R.x<>0 by FDIFF_1:26
,SIN_COS:54,TAYLOR_1:16;
  then
A7: (exp_R*f)/exp_R is_differentiable_on Z by A6,FDIFF_2:21;
A8: log(number_e,a)-1<>0
  proof
A9: number_e<>1 by TAYLOR_1:11;
    assume log(number_e,a)-1=0;
    then log(number_e,a)=log(number_e,number_e) by A9,POWER:52,TAYLOR_1:11;
    then a=(number_e) to_power log(number_e,number_e) by A3,A9,POWER:def 3
,TAYLOR_1:11
      .=number_e by A9,POWER:def 3,TAYLOR_1:11;
    hence contradiction by A4;
  end;
  for x st x in Z holds (((1/(log(number_e,a)-1))(#)((exp_R*f)/exp_R))`|Z
  ).x =a #R x/exp_R.x
  proof
    let x;
A10: exp_R.x <>0 by SIN_COS:54;
    assume
A11: x in Z;
    then
A12: (exp_R*f).x=exp_R.(f.x) by A5,FUNCT_1:12
      .=exp_R.(x*log(number_e,a)) by A2,A11
      .=a #R x by A3,Th1;
A13: exp_R is_differentiable_in x & exp_R*f is_differentiable_in x by A6,A11,
FDIFF_1:9,SIN_COS:65;
    (((1/(log(number_e,a)-1))(#)((exp_R*f)/exp_R))`|Z).x =(1/(log(
    number_e,a)-1))*diff(((exp_R*f)/exp_R),x) by A1,A7,A11,FDIFF_1:20
      .=(1/(log(number_e,a)-1))* ((diff(exp_R*f,x) * exp_R.x - diff(exp_R,x)
    *(exp_R*f).x)/(exp_R.x)^2) by A13,A10,FDIFF_2:14
      .=(1/(log(number_e,a)-1))*((diff(exp_R*f,x)* exp_R.x-exp_R.x*a #R x)/(
    exp_R.x)^2) by A12,SIN_COS:65
      .=(1/(log(number_e,a)-1))*(((diff(exp_R*f,x)-a #R x)*exp_R.x)/ ((exp_R
    .x)*(exp_R.x)))
      .=(1/(log(number_e,a)-1))*((diff(exp_R*f,x)-a #R x)/(exp_R.x)) by A10,
XCMPLX_1:91
      .=(1/(log(number_e,a)-1))*(diff(exp_R*f,x)-a #R x)/(exp_R.x) by
XCMPLX_1:74
      .=(1/(log(number_e,a)-1))*(((exp_R*f)`|Z).x-a #R x)/(exp_R.x) by A6,A11,
FDIFF_1:def 7
      .=(1/(log(number_e,a)-1))*(a #R x*log(number_e,a)-a #R x)/(exp_R.x) by A2
,A3,A5,A11,Th11
      .=(1/(log(number_e,a)-1))*(log(number_e,a)-1)*a #R x/exp_R.x
      .=1* a #R x/exp_R.x by A8,XCMPLX_1:106
      .=a #R x/exp_R.x;
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:20;
end;
