reserve y for set,
  x,r,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,f3 for PartFunc of REAL,REAL;

theorem Th27:
  Z c= dom (( #R (1/2))*f) & f=f1-f2 & f2=#Z 2 & (for x st x in Z
  holds f1.x=a^2 & f.x >0) implies ( #R (1/2))*f is_differentiable_on Z & for x
  st x in Z holds ((( #R (1/2))*f)`|Z).x = -x*(a^2-x #Z 2) #R (-1/2)
proof
  assume that
A1: Z c= dom (( #R (1/2))*f) and
A2: f=f1-f2 and
A3: f2=#Z 2 and
A4: for x st x in Z holds f1.x=a^2 & f.x >0;
  for y being object st y in Z holds y in dom f by A1,FUNCT_1:11;
  then
A5: Z c= dom ((f1+(-1)(#)f2)) by A2,TARSKI:def 3;
A6: for x st x in Z holds f1.x=a^2+0*x by A4;
  then
A7: f is_differentiable_on Z by A2,A3,A5,FDIFF_4:12;
A8: for x st x in Z holds ( #R (1/2))*f is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then f is_differentiable_in x & f.x >0 by A4,A7,FDIFF_1:9;
    hence thesis by TAYLOR_1:22;
  end;
  then
A9: ( #R (1/2))*f is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((( #R (1/2))*f)`|Z).x = -x*(a^2-x #Z 2) #R (-1/2 )
  proof
    let x;
    assume
A10: x in Z;
    then
A11: f is_differentiable_in x & f.x >0 by A4,A7,FDIFF_1:9;
    x in dom (f1-f2) by A1,A2,A10,FUNCT_1:11;
    then
A12: (f1-f2).x=f1.x - f2.x by VALUED_1:13
      .=a^2 -(f2.x) by A4,A10
      .=a^2 -(x #Z 2) by A3,TAYLOR_1:def 1;
    ((( #R (1/2))*f)`|Z).x = diff(( #R (1/2))*f,x) by A9,A10,FDIFF_1:def 7
      .=(1/2)*((f.x) #R (1/2-1)) * diff(f,x) by A11,TAYLOR_1:22
      .=(1/2)*((f.x) #R (1/2-1))*(f`|Z).x by A7,A10,FDIFF_1:def 7
      .=(1/2)*((f.x) #R (1/2-1))*(0+2*(-1)*x) by A2,A3,A5,A6,A10,FDIFF_4:12
      .=-x*(a^2-x #Z 2) #R (-1/2) by A2,A12;
    hence thesis;
  end;
  hence thesis by A1,A8,FDIFF_1:9;
end;
