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 -(( #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 |^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;
A5: for x st x in Z holds f1.x=a^2+0*x by A4;
A6: Z c= dom (( #R (1/2))*f) by A1,VALUED_1:8;
  then for y being object st y in Z holds y in dom f by FUNCT_1:11;
  then
A7: Z c= dom ((f1+(-1)(#)f2)) by A2,TARSKI:def 3;
  then
A8: f is_differentiable_on Z by A2,A3,A5,Th12;
  now
    let x;
    assume x in Z;
    then f is_differentiable_in x & f.x >0 by A4,A8,FDIFF_1:9;
    hence ( #R (1/2))*f is_differentiable_in x by TAYLOR_1:22;
  end;
  then
A9: ( #R (1/2))*f is_differentiable_on Z by A6,FDIFF_1:9;
  for x st x in Z holds (((-1)(#)(( #R (1/2))*f))`|Z).x =x* (a^2-x |^2)
  #R (-1/2)
  proof
    let x;
    assume
A10: x in Z;
    then
A11: f is_differentiable_in x & f.x >0 by A4,A8,FDIFF_1:9;
    x in dom (f1-f2) by A2,A6,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
      .=a^2 -(x |^2) by PREPOWER:36;
    (((-1)(#)(( #R (1/2))*f))`|Z).x =(-1)*diff((( #R (1/2))*f),x) by A1,A9,A10,
FDIFF_1:20
      .=(-1)*((1/2)*( ( f.x) #R (1/2-1)) * diff(f,x)) by A11,TAYLOR_1:22
      .=(-1)*((1/2)*( ( f.x) #R (1/2-1))*(f`|Z).x) by A8,A10,FDIFF_1:def 7
      .=(-1)*((1/2)*( ( f.x) #R (1/2-1))*(0+2*(-1)*x)) by A2,A3,A7,A5,A10,Th12
      .=x* (a^2-x |^2) #R (-1/2) by A2,A12;
    hence thesis;
  end;
  hence thesis by A1,A9,FDIFF_1:20;
end;
