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/(exp_R*f))) & (for x st x
  in Z holds f.x=x*log(number_e,a)) & a>0 & a<>number_e implies (1/(1-log(
  number_e,a)))(#)(exp_R/(exp_R*f)) is_differentiable_on Z & for x st x in Z
  holds (((1/(1-log(number_e,a)))(#)(exp_R/(exp_R*f)))`|Z).x =exp_R.x/a #R x
proof
  assume that
A1: Z c= dom ((1/(1-log(number_e,a)))(#)(exp_R/(exp_R*f))) 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/(exp_R*f)) by A1,VALUED_1:def 5;
  then Z c= dom exp_R /\ (dom (exp_R*f) \ (exp_R*f)"{0}) by RFUNCT_1:def 1;
  then
A5: Z c= dom (exp_R*f) by XBOOLE_1:1;
  then
A6: exp_R*f is_differentiable_on Z by A2,A3,Th11;
A7: for x st x in Z holds (exp_R*f).x<>0
  proof
    let x;
    assume x in Z;
    then (exp_R*f).x=exp_R.(f.x) by A5,FUNCT_1:12;
    hence thesis by SIN_COS:54;
  end;
  exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  then
A8: exp_R/(exp_R*f) is_differentiable_on Z by A6,A7,FDIFF_2:21;
A9: 1-log(number_e,a)<>0
  proof
A10: number_e<>1 by TAYLOR_1:11;
    assume 1-log(number_e,a)=0;
    then log(number_e,a)=log(number_e,number_e) by A10,POWER:52,TAYLOR_1:11;
    then a=(number_e) to_power log(number_e,number_e) by A3,A10,POWER:def 3
,TAYLOR_1:11
      .=number_e by A10,POWER:def 3,TAYLOR_1:11;
    hence contradiction by A4;
  end;
  for x st x in Z holds (((1/(1-log(number_e,a)))(#)(exp_R/(exp_R*f)))`|Z
  ).x =exp_R.x/a #R x
  proof
    let x;
A11: exp_R is_differentiable_in x by SIN_COS:65;
A12: a #R x >0 by A3,PREPOWER:81;
    assume
A13: x in Z;
    then
A14: (exp_R*f).x=exp_R.(f.x) by A5,FUNCT_1:12
      .=exp_R.(x*log(number_e,a)) by A2,A13
      .=a #R x by A3,Th1;
A15: exp_R*f is_differentiable_in x & (exp_R*f).x <>0 by A6,A7,A13,FDIFF_1:9;
    (((1/(1-log(number_e,a)))(#)(exp_R/(exp_R*f)))`|Z).x =(1/(1-log(
    number_e,a)))*diff((exp_R/(exp_R*f)),x) by A1,A8,A13,FDIFF_1:20
      .=(1/(1-log(number_e,a)))* ((diff(exp_R,x) * (exp_R*f).x - diff((exp_R
    *f),x) * exp_R.x)/((exp_R*f).x)^2) by A11,A15,FDIFF_2:14
      .=(1/(1-log(number_e,a)))* ((exp_R.x * a #R x - diff((exp_R*f),x)*
    exp_R.x)/(a #R x)^2) by A14,SIN_COS:65
      .=(1/(1-log(number_e,a)))*((exp_R.x*(a #R x -diff((exp_R*f),x)))/(a #R
    x)^2)
      .=(1/(1-log(number_e,a)))*((exp_R.x*(a #R x - ((exp_R*f)`|Z).x))/(a #R
    x)^2) by A6,A13,FDIFF_1:def 7
      .=(1/(1-log(number_e,a)))* ((exp_R.x*(a #R x -a #R x*log(number_e,a)))
    /(a #R x)^2) by A2,A3,A5,A13,Th11
      .=(1/(1-log(number_e,a)))*((1-log(number_e,a))*exp_R.x*a #R x)/ (a #R
    x)^2 by XCMPLX_1:74
      .=(1/(1-log(number_e,a)))*(1-log(number_e,a))*exp_R.x*a #R x/(a #R x)
    ^2
      .=1*exp_R.x*a #R x/(a #R x)^2 by A9,XCMPLX_1:106
      .=exp_R.x/a #R x by A12,XCMPLX_1:91;
    hence thesis;
  end;
  hence thesis by A1,A8,FDIFF_1:20;
end;
