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