reserve y for set;
reserve x,a,b,c for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom ((-2/3)(#)(( #R (3/2))*f)) & (for x st x in Z holds f.x=a-x &
  f.x> 0 ) implies (-2/3)(#)(( #R (3/2))*f) is_differentiable_on Z & for x st x
  in Z holds (((-2/3)(#)(( #R (3/2))*f))`|Z).x =(a-x) #R (1/2)
proof
  assume that
A1: Z c= dom ((-2/3)(#)(( #R (3/2))*f)) and
A2: for x st x in Z holds f.x=a-x & f.x>0;
A3: Z c= dom (( #R (3/2))*f) by A1,VALUED_1:def 5;
  then for y being object st y in Z holds y in dom f by FUNCT_1:11;
  then
A4: Z c= dom f by TARSKI:def 3;
A5: for x st x in Z holds f.x =(-1)*x+a
  proof
    let x;
    assume x in Z;
    then f.x =a-x by A2;
    hence thesis;
  end;
  then
A6: f is_differentiable_on Z by A4,FDIFF_1:23;
  now
    let x;
    assume x in Z;
    then f is_differentiable_in x & f.x >0 by A2,A6,FDIFF_1:9;
    hence ( #R (3/2))*f is_differentiable_in x by TAYLOR_1:22;
  end;
  then
A7: ( #R (3/2))*f is_differentiable_on Z by A3,FDIFF_1:9;
  for x st x in Z holds (((-2/3)(#)(( #R (3/2))*f))`|Z).x =(a-x) #R (1/2)
  proof
    let x;
    assume
A8: x in Z;
    then
A9: f.x = a-x by A2;
A10: f is_differentiable_in x & f.x >0 by A2,A6,A8,FDIFF_1:9;
    (((-2/3)(#)(( #R (3/2))*f))`|Z).x =(-2/3)*diff((( #R (3/2))*f),x) by A1,A7
,A8,FDIFF_1:20
      .=(-2/3)* ((3/2)*( ( f.x) #R (3/2-1)) * diff(f,x)) by A10,TAYLOR_1:22
      .=(-2/3)* ((3/2)*( ( f.x) #R (3/2-1))*(f`|Z).x) by A6,A8,FDIFF_1:def 7
      .=(-2/3)* ((3/2)*( (a-x) #R (3/2-1))*(-1)) by A4,A5,A8,A9,FDIFF_1:23
      .=(a-x) #R (1/2);
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:20;
end;
