reserve y for set,
  x,a for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2 for PartFunc of REAL,REAL;

theorem Th43:
  Z c= dom ((1/(4*a))(#)(sin*f)) & (for x st x in Z holds f.x=2*a*
x) & a<>0 implies (1/(4*a))(#)(sin*f) is_differentiable_on Z & for x st x in Z
  holds (((1/(4*a))(#)(sin*f))`|Z).x =1/2 * cos(2*a*x)
proof
  assume that
A1: Z c= dom ((1/(4*a))(#)(sin*f)) and
A2: for x st x in Z holds f.x=2*a*x and
A3: a<>0;
A4: Z c= dom (sin*f) & for x st x in Z holds f.x=(2*a)*x+0 by A1,A2,
VALUED_1:def 5;
  then
A5: sin*f is_differentiable_on Z by FDIFF_4:37;
  for x st x in Z holds (((1/(4*a))(#)(sin*f))`|Z).x =1/2 * cos(2*a*x)
  proof
    let x;
    assume
A6: x in Z;
    then (((1/(4*a))(#)(sin*f))`|Z).x =(1/(4*a))*diff((sin*f),x) by A1,A5,
FDIFF_1:20
      .=(1/(4*a))*((sin*f)`|Z).x by A5,A6,FDIFF_1:def 7
      .=(1/(4*a))*((2*a)* cos.((2*a)*x+0)) by A4,A6,FDIFF_4:37
      .=1/(4*a)*(2*a)* cos.((2*a)*x+0)
      .=(1/4)*(1/a)*(2*a)* cos.((2*a)*x) by XCMPLX_1:102
      .=(1/4)*2*(1/a*a)* cos.(2*a*x)
      .=(1/2)*1*cos.(2*a*x) by A3,XCMPLX_1:106
      .=1/2*cos(2*a*x) by SIN_COS:def 19;
    hence thesis;
  end;
  hence thesis by A1,A5,FDIFF_1:20;
end;
