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 Th11:
  Z c= dom (exp_R*f) & (for x st x in Z holds f.x=x*log(number_e,a
)) & a>0 implies exp_R*f is_differentiable_on Z & for x st x in Z holds ((exp_R
  *f)`|Z).x =a #R x*log(number_e,a)
proof
  assume that
A1: Z c= dom (exp_R*f) and
A2: for x st x in Z holds f.x=x*log(number_e,a) and
A3: a>0;
  for y being object
st y in Z holds y in dom f by A1,FUNCT_1:11;
  then
A4: Z c= dom f by TARSKI:def 3;
A5: for x st x in Z holds f.x = log(number_e,a)*x+0 by A2;
  then
A6: f is_differentiable_on Z by A4,FDIFF_1:23;
A7: for x st x in Z holds exp_R*f is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then f is_differentiable_in x by A6,FDIFF_1:9;
    hence thesis by TAYLOR_1:19;
  end;
  then
A8: exp_R*f is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((exp_R*f)`|Z).x = a #R x*log(number_e,a)
  proof
    let x;
    assume
A9: x in Z;
    then f is_differentiable_in x by A6,FDIFF_1:9;
    then diff(exp_R*f,x) = exp_R.(f.x)*diff(f,x) by TAYLOR_1:19
      .=exp_R.(f.x)*(f`|Z).x by A6,A9,FDIFF_1:def 7
      .=exp_R.(f.x)*(log(number_e,a)) by A4,A5,A9,FDIFF_1:23
      .=exp_R.(x*log(number_e,a))*(log(number_e,a)) by A2,A9
      .=a #R x*log(number_e,a) by A3,Th1;
    hence thesis by A8,A9,FDIFF_1:def 7;
  end;
  hence thesis by A1,A7,FDIFF_1:9;
end;
