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 Th49:
  Z c= dom ((1/2)(#)(( #Z 2)*sin)) implies (1/2)(#)(( #Z 2)*sin)
is_differentiable_on Z & for x st x in Z holds (((1/2)(#)(( #Z 2)*sin))`|Z).x =
  sin.x*cos.x
proof
A1: now
    let x;
    assume x in Z;
    sin is_differentiable_in x by SIN_COS:64;
    hence ( #Z 2)*sin is_differentiable_in x by TAYLOR_1:3;
  end;
  assume
A2: Z c= dom ((1/2)(#)(( #Z 2)*sin));
  then Z c= dom (( #Z 2)*sin) by VALUED_1:def 5;
  then
A3: ( #Z 2)*sin is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds (((1/2)(#)(( #Z 2)*sin))`|Z).x =sin.x*cos.x
  proof
    let x;
A4: sin is_differentiable_in x by SIN_COS:64;
    assume x in Z;
    then
    (((1/2)(#)(( #Z 2)*sin))`|Z).x =(1/2)*diff((( #Z 2)*sin),x) by A2,A3,
FDIFF_1:20
      .=(1/2)*(2*( (sin.x) #Z (2-1)) * diff(sin,x)) by A4,TAYLOR_1:3
      .=(1/2)*(2*( (sin.x) #Z (2-1)) *cos.x) by SIN_COS:64
      .=( (sin.x) #Z (2-1)) *cos.x
      .=sin.x*cos.x by PREPOWER:35;
    hence thesis;
  end;
  hence thesis by A2,A3,FDIFF_1:20;
end;
