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 ((2/(3*log(number_e,a)))(#)(( #R (3/2))*(f+(exp_R*f1)))) & (
for x st x in Z holds f.x=1 & f1.x=x*log(number_e,a)) & a>0 & a<>1 implies (2/(
3*log(number_e,a)))(#)(( #R (3/2))*(f+(exp_R*f1))) is_differentiable_on Z & for
x st x in Z holds (((2/(3*log(number_e,a)))(#)(( #R (3/2))*(f+(exp_R*f1))))`|Z)
  .x =a #R x * (1+a #R x) #R (1/2)
proof
  assume that
A1: Z c= dom ((2/(3*log(number_e,a)))(#)(( #R (3/2))*(f+(exp_R*f1)))) and
A2: for x st x in Z holds f.x=1 & f1.x=x*log(number_e,a) and
A3: a>0 and
A4: a<>1;
A5: for x st x in Z holds f.x=0*x+1 by A2;
A6: Z c= dom (( #R (3/2))*(f+(exp_R*f1))) by A1,VALUED_1:def 5;
  then for y being object
st y in Z holds y in dom (f+(exp_R*f1)) by FUNCT_1:11;
  then
A7: Z c= dom (f+(exp_R*f1)) by TARSKI:def 3;
  then
A8: Z c= dom (exp_R*f1) /\ dom f by VALUED_1:def 1;
  then
A9: Z c= dom (exp_R*f1) by XBOOLE_1:18;
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,A9,Th11;
A12: Z c= dom f by A8,XBOOLE_1:18;
  then
A13: f is_differentiable_on Z by A5,FDIFF_1:23;
  then
A14: f+(exp_R*f1) is_differentiable_on Z by A7,A11,FDIFF_1:18;
A15: for x st x in Z holds (f+(exp_R*f1)).x>0
  proof
    let x;
    assume
A16: x in Z;
    then (f+(exp_R*f1)).x=f.x+(exp_R*f1).x by A7,VALUED_1:def 1
      .=f.x+exp_R.(f1.x) by A9,A16,FUNCT_1:12
      .=1+exp_R.(f1.x) by A2,A16
      .=1+exp_R.(x*log(number_e,a)) by A2,A16;
    hence thesis by SIN_COS:54,XREAL_1:34;
  end;
  now
    let x;
    assume x in Z;
    then f+(exp_R*f1) is_differentiable_in x & (f+(exp_R*f1)).x>0 by A14,A15,
FDIFF_1:9;
    hence ( #R (3/2))*(f+(exp_R*f1)) is_differentiable_in x by TAYLOR_1:22;
  end;
  then
A17: ( #R (3/2))*(f+(exp_R*f1)) is_differentiable_on Z by A6,FDIFF_1:9;
A18: log(number_e,a)<>0
  proof
A19: 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,A19,POWER:def 3
,TAYLOR_1:11
      .=1 by A19,POWER:def 3,TAYLOR_1:11;
    hence contradiction by A4;
  end;
  for x st x in Z holds (((2/(3*log(number_e,a)))(#)(( #R (3/2))*(f+(
  exp_R*f1))))`|Z).x =a #R x * (1+a #R x) #R (1/2)
  proof
    let x;
A20: 3*log(number_e,a) <> 0 by A18;
    assume
A21: x in Z;
    then
A22: ((f+(exp_R*f1))`|Z).x=diff(f,x)+diff((exp_R*f1),x) by A7,A13,A11,
FDIFF_1:18
      .=diff(f,x)+((exp_R*f1)`|Z).x by A11,A21,FDIFF_1:def 7
      .=(f`|Z).x+((exp_R*f1)`|Z).x by A13,A21,FDIFF_1:def 7
      .=0+((exp_R*f1)`|Z).x by A12,A5,A21,FDIFF_1:23
      .=a #R x*log(number_e,a) by A3,A10,A9,A21,Th11;
A23: (f+(exp_R*f1)).x=f.x+(exp_R*f1).x by A7,A21,VALUED_1:def 1
      .=f.x+exp_R.(f1.x) by A9,A21,FUNCT_1:12
      .=1+exp_R.(f1.x) by A2,A21
      .=1+exp_R.(x*log(number_e,a)) by A2,A21
      .=1+a #R x by A3,Th1;
    f+(exp_R*f1) is_differentiable_in x & (f+(exp_R*f1)).x>0 by A14,A15,A21,
FDIFF_1:9;
    then
    diff((( #R (3/2))*(f+(exp_R*f1))),x) =(3/2)*(((f+(exp_R*f1)).x) #R (3
    /2-1)) * diff((f+(exp_R*f1)),x) by TAYLOR_1:22
      .=(3/2)*((1+a #R x) #R (1/2)) *(a #R x*log(number_e,a)) by A14,A21,A23
,A22,FDIFF_1:def 7
      .=(3*log(number_e,a))/2*a #R x*(1+a #R x) #R (1/2);
    then
    (((2/(3*log(number_e,a)))(#)(( #R (3/2))*(f+(exp_R*f1))))`|Z).x =(2/(
3*log(number_e,a)))*((3*log(number_e,a))/2*a #R x*(1+a #R x) #R (1/2)) by A1
,A17,A21,FDIFF_1:20
      .=(2/(3*log(number_e,a)))*((3*log(number_e,a))/2) *a #R x*(1+a #R x)
    #R (1/2)
      .=1*a #R x*(1+a #R x) #R (1/2) by A20,XCMPLX_1:112
      .=a #R x*(1+a #R x) #R (1/2);
    hence thesis;
  end;
  hence thesis by A1,A17,FDIFF_1:20;
end;
