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))*sin))) & (for x st x in Z holds sin.x>0)
implies 2(#)(( #R (1/2))*sin) is_differentiable_on Z & for x st x in Z holds ((
  2(#)(( #R (1/2))*sin))`|Z).x =cos.x*(sin.x) #R (-1/2)
proof
  assume that
A1: Z c= dom ((2(#)(( #R (1/2))*sin))) and
A2: for x st x in Z holds sin.x>0;
A3: now
    let x;
    assume x in Z;
    then sin is_differentiable_in x & sin.x >0 by A2,SIN_COS:64;
    hence ( #R (1/2))*sin is_differentiable_in x by TAYLOR_1:22;
  end;
  Z c= dom (( #R (1/2))*sin) by A1,VALUED_1:def 5;
  then
A4: ( #R (1/2))*sin is_differentiable_on Z by A3,FDIFF_1:9;
  for x st x in Z holds ((2(#)(( #R (1/2))*sin))`|Z).x =cos.x*(sin.x) #R (-1/2)
  proof
    let x;
    assume
A5: x in Z;
    then
A6: sin is_differentiable_in x & sin.x >0 by A2,SIN_COS:64;
    ((2(#)(( #R (1/2))*sin))`|Z).x =2*diff((( #R (1/2))*sin),x) by A1,A4,A5,
FDIFF_1:20
      .=2*((1/2)*( (sin.x) #R (1/2-1)) * diff(sin,x)) by A6,TAYLOR_1:22
      .=2*((1/2)*( (sin.x) #R (1/2-1)) *cos.x) by SIN_COS:64
      .=cos.x*(sin.x) #R (-1/2);
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:20;
end;
