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
A4: (( #R (3/2))*f) is_differentiable_on Z by A2,Th27;
  for x st x in Z holds (((2/3)(#)(( #R (3/2))*f))`|Z).x =(a+x) #R (1/2)
  proof
    let x;
    assume
A5: x in Z;
    hence
    (((2/3)(#)(( #R (3/2))*f))`|Z).x =(2/3)*diff((( #R (3/2))*f),x) by A1,A4,
FDIFF_1:20
      .=(2/3)*((( #R (3/2))*f)`|Z).x by A4,A5,FDIFF_1:def 7
      .=(2/3)*((3/2)* (a+x) #R (1/2)) by A2,A3,A5,Th27
      .=(a+x) #R (1/2);
  end;
  hence thesis by A1,A4,FDIFF_1:20;
end;
