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 Th27:
  Z c= dom (( #R (3/2))*f) & (for x st x in Z holds f.x=a+x & f.x>
0) implies ( #R (3/2))*f is_differentiable_on Z & for x st x in Z holds ((( #R
  (3/2))*f)`|Z).x =(3/2)* (a+x) #R (1/2)
proof
  assume that
A1: Z c= dom (( #R (3/2))*f) and
A2: for x st x in Z holds f.x=a+x & f.x>0;
  for y being object st y in Z holds y in dom f by A1,FUNCT_1:11;
  then
A3: Z c= dom f by TARSKI:def 3;
A4: for x st x in Z holds f.x = 1*x+a by A2;
  then
A5: f is_differentiable_on Z by A3,FDIFF_1:23;
A6: for x st x in Z holds ( #R (3/2))*f is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then f is_differentiable_in x & f.x >0 by A2,A5,FDIFF_1:9;
    hence thesis by TAYLOR_1:22;
  end;
  then
A7: ( #R (3/2))*f is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((( #R (3/2))*f)`|Z).x =(3/2)*(a+x) #R (1/2)
  proof
    let x;
    assume
A8: x in Z;
    then
A9: f.x = a+x by A2;
    f is_differentiable_in x & f.x >0 by A2,A5,A8,FDIFF_1:9;
    then diff(( #R (3/2)) *f,x) = (3/2)*( ( f.x) #R (3/2-1)) * diff(f,x) by
TAYLOR_1:22
      .= (3/2)*( ( f.x) #R (3/2-1))*(f`|Z).x by A5,A8,FDIFF_1:def 7
      .= (3/2)*( (a+x) #R (3/2-1))*1 by A3,A4,A8,A9,FDIFF_1:23
      .=(3/2)* (a+x) #R (1/2);
    hence thesis by A7,A8,FDIFF_1:def 7;
  end;
  hence thesis by A1,A6,FDIFF_1:9;
end;
