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*b))(#)(( #R (3/2))*f)) & (for x st x in Z holds f.x=a
-b*x & b<>0 & f.x>0) implies (-2/(3*b))(#)(( #R (3/2))*f) is_differentiable_on
Z & for x st x in Z holds (((-2/(3*b))(#)(( #R (3/2))*f))`|Z).x =(a-b*x) #R (1/
  2)
proof
  assume that
A1: Z c= dom ((-2/(3*b))(#)(( #R (3/2))*f)) and
A2: for x st x in Z holds f.x=a-b*x & b<>0 & 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 =(-b)*x+a
  proof
    let x;
    assume x in Z;
    then f.x =a-b*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*b))(#)(( #R (3/2))*f))`|Z).x =(a-b*x) #R (1/2)
  proof
    let x;
    assume
A8: x in Z;
    then
A9: 3*b <>0 by A2;
A10: f.x = a-b*x by A2,A8;
A11: f is_differentiable_in x & f.x >0 by A2,A6,A8,FDIFF_1:9;
    (((-2/(3*b))(#)(( #R (3/2))*f))`|Z).x =(-2/(3*b))*diff((( #R (3/2))*f
    ), x ) by A1,A7,A8,FDIFF_1:20
      .=(-2/(3*b))* ((3/2)*( ( f.x) #R (3/2-1)) * diff(f,x)) by A11,TAYLOR_1:22
      .=(-2/(3*b))* ((3/2)*( ( f.x) #R (3/2-1))*(f`|Z).x) by A6,A8,
FDIFF_1:def 7
      .=(-2/(3*b))* ((3/2)*( (a-b*x) #R (3/2-1))*(-b)) by A4,A5,A8,A10,
FDIFF_1:23
      .=(2/(3*b))* ((3*b)/2)*( (a-b*x) #R (3/2-1))
      .=1*( (a-b*x) #R (3/2-1)) by A9,XCMPLX_1:112
      .=(a-b*x) #R (1/2);
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:20;
end;
