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))(#)((exp_R*f1)(#)f2)) & (for x st x in Z
holds f1.x=x*log(number_e,a) & f2.x=x-1/log(number_e,a)) & a>0 & a<>1 implies (
1/log(number_e,a))(#)((exp_R*f1)(#)f2) is_differentiable_on Z & for x st x in Z
  holds (((1/log(number_e,a))(#)((exp_R*f1)(#)f2))`|Z).x =x * a #R x
proof
  assume that
A1: Z c= dom ((1/log(number_e,a))(#)((exp_R*f1)(#)f2)) and
A2: for x st x in Z holds f1.x=x*log(number_e,a) & f2.x=x-1/log(
  number_e,a) and
A3: a>0 and
A4: a<>1;
A5: Z c= dom ((exp_R*f1)(#)f2) by A1,VALUED_1:def 5;
  then
A6: Z c= dom (exp_R*f1) /\ dom f2 by VALUED_1:def 4;
  then
A7: Z c= dom (exp_R*f1) by XBOOLE_1:18;
A8: for x st x in Z holds f2.x = 1*x+-1/log(number_e,a)
  proof
    let x;
A9: 1*x+-1/log(number_e,a)=1*x-1/log(number_e,a);
    assume x in Z;
    hence thesis by A2,A9;
  end;
A10: for x st x in Z holds f1.x=x*log(number_e,a) by A2;
  then
A11: exp_R*f1 is_differentiable_on Z by A3,A7,Th11;
A12: Z c= dom f2 by A6,XBOOLE_1:18;
  then
A13: f2 is_differentiable_on Z by A8,FDIFF_1:23;
  then
A14: ((exp_R*f1)(#)f2) is_differentiable_on Z by A5,A11,FDIFF_1:21;
A15: log(number_e,a)<>0
  proof
A16: number_e<>1 by TAYLOR_1:11;
    assume log(number_e,a)=0;
    then log(number_e,a)=log(number_e,1) by SIN_COS2:13,TAYLOR_1:13;
    then a=(number_e) to_power log(number_e,1) by A3,A16,POWER:def 3
,TAYLOR_1:11
      .=1 by A16,POWER:def 3,TAYLOR_1:11;
    hence contradiction by A4;
  end;
  for x st x in Z holds (((1/log(number_e,a))(#)((exp_R*f1)(#)f2))`|Z).x
  =x * a #R x
  proof
    let x;
    assume
A17: x in Z;
    then
A18: (exp_R*f1).x=exp_R.(f1.x) by A7,FUNCT_1:12
      .=exp_R.(x*log(number_e,a)) by A2,A17
      .=a #R x by A3,Th1;
    (((1/log(number_e,a))(#)((exp_R*f1)(#)f2))`|Z).x =(1/log(number_e,a))
    *diff((exp_R*f1)(#)f2,x) by A1,A14,A17,FDIFF_1:20
      .=(1/log(number_e,a))*(((exp_R*f1)(#)f2)`|Z).x by A14,A17,FDIFF_1:def 7
      .=(1/log(number_e,a))*(f2.x*diff(exp_R*f1,x)+(exp_R*f1).x*diff(f2,x))
    by A5,A11,A13,A17,FDIFF_1:21
      .=(1/log(number_e,a))*(f2.x*((exp_R*f1)`|Z).x+(exp_R*f1).x*diff(f2,x))
    by A11,A17,FDIFF_1:def 7
      .=(1/log(number_e,a))*(f2.x*((exp_R*f1)`|Z).x+(exp_R*f1).x*(f2`|Z).x)
    by A13,A17,FDIFF_1:def 7
      .=(1/log(number_e,a))*(f2.x*(a #R x*log(number_e,a)) +(exp_R*f1).x*(f2
    `|Z).x) by A3,A10,A7,A17,Th11
      .=(1/log(number_e,a))*(f2.x*(a#R x*log(number_e,a))+(exp_R*f1).x*1) by
A12,A8,A17,FDIFF_1:23
      .=(1/log(number_e,a))*((f2.x*log(number_e,a)+1)*a #R x) by A18
      .=(1/log(number_e,a))*(((x-1/log(number_e,a))* log(number_e,a)+1)*a #R
    x) by A2,A17
      .=(1/log(number_e,a))*(x*log(number_e,a) -1/log(number_e,a)*log(
    number_e,a)+1)*a #R x
      .=(1/log(number_e,a))*(x*log(number_e,a)-1+1)*a #R x by A15,XCMPLX_1:106
      .=1/log(number_e,a)*log(number_e,a)*x*a #R x
      .=1*x*a #R x by A15,XCMPLX_1:106
      .=x*a #R x;
    hence thesis;
  end;
  hence thesis by A1,A14,FDIFF_1:20;
end;
