reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  for x holds bD(sin(#)cos(#)cos,h).x = (1/2)*(cos((2*x-h)/2)*sin(h/2)+
  cos((6*x-3*h)/2)*sin(3*h/2))
proof
  let x;
  set y=3*x;
  set z=3*h;
  bD(sin(#)cos(#)cos,h).x = (sin(#)cos(#)cos).x -(sin(#)cos(#)cos).(x-h)
  by DIFF_1:4
    .= ((sin(#)cos).x)*(cos.x) -(sin(#)cos(#)cos).(x-h) by VALUED_1:5
    .= (sin.x)*(cos.x)*(cos.x) -(sin(#)cos(#)cos).(x-h) by VALUED_1:5
    .= (sin.x)*(cos.x)*(cos.x) -((sin(#)cos).(x-h))*(cos.(x-h)) by VALUED_1:5
    .= sin(x)*cos(x)*cos(x) -sin(x-h)*cos(x-h)*cos(x-h) by VALUED_1:5
    .= (1/4)*(sin(x+x-x)-sin(x+x-x)+sin(x+x-x)+sin(x+x+x)) -sin(x-h)*cos(x-h
  )*cos(x-h) by SIN_COS4:35
    .= (1/4)*(sin(x)+sin(3*x))-(1/4)*(sin((x-h)+(x-h)-(x-h)) -sin((x-h)+(x-h
  )-(x-h))+sin((x-h)+(x-h)-(x-h)) +sin((x-h)+(x-h)+(x-h))) by SIN_COS4:35
    .= (1/4)*(sin(x)-sin(x-h))+(1/4)*(sin(3*x)-sin(3*(x-h)))
    .= (1/4)*(2*(cos((x+(x-h))/2)*sin((x-(x-h))/2))) +(1/4)*(sin(3*x)-sin(3*
  (x-h))) by SIN_COS4:16
    .= (1/4)*(2*(cos((2*x-h)/2)*sin(h/2)))+(1/4) *(2*(cos((y+(y-z))/2)*sin((
  y-(y-z))/2))) by SIN_COS4:16
    .= (1/2)*(cos((2*x-h)/2)*sin(h/2)) +(1/2)*((cos((6*x-3*h)/2)*sin(3*h/2))
  );
  hence thesis;
end;
