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(#)(( #R (1/2))*f))) & (for x st x in Z holds f.x=a+x & f.
x>0) implies 2(#)(( #R (1/2))*f) is_differentiable_on Z & for x st x in Z holds
  ((2(#)(( #R (1/2))*f))`|Z).x =(a+x) #R (-1/2)
proof
  assume that
A1: Z c= dom ((2(#)(( #R (1/2))*f))) and
A2: for x st x in Z holds f.x=a+x & f.x>0;
A3: Z c= dom (( #R (1/2))*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 by A2;
  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 (1/2))*f is_differentiable_in x by TAYLOR_1:22;
  end;
  then
A7: ( #R (1/2))*f is_differentiable_on Z by A3,FDIFF_1:9;
  for x st x in Z holds ((2(#)(( #R (1/2))*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(#)(( #R (1/2))*f))`|Z).x =2*diff((( #R (1/2))*f),x) by A1,A7,A8,
FDIFF_1:20
      .=2*((1/2)*( ( f.x) #R (1/2-1)) * diff(f,x)) by A10,TAYLOR_1:22
      .=2*((1/2)*( ( f.x) #R (1/2-1))*(f`|Z).x) by A6,A8,FDIFF_1:def 7
      .=2*((1/2)*( (a+x) #R (1/2-1))*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;
