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(#)sin(#)cos,h).x = -(1/2)*(sin(x)*sin(h/2))+(1/2)*(
  sin(3*x)*sin(3*h/2))
proof
  let x;
  set y=3*x;
  set z=3*h;
  cD(sin(#)sin(#)cos,h).x = (sin(#)sin(#)cos).(x+h/2) - (sin(#)sin(#)cos).
  (x-h/2) by DIFF_1:5
    .= ((sin(#)sin).(x+h/2))*(cos.(x+h/2)) -(sin(#)sin(#)cos).(x-h/2) by
VALUED_1:5
    .= (sin.(x+h/2))*(sin.(x+h/2))*(cos.(x+h/2)) -(sin(#)sin(#)cos).(x-h/2)
  by VALUED_1:5
    .= (sin.(x+h/2))*(sin.(x+h/2))*(cos.(x+h/2)) -((sin(#)sin).(x-h/2))*(cos
  .(x-h/2)) by VALUED_1:5
    .= sin(x+h/2)*sin(x+h/2)*cos(x+h/2) -sin(x-h/2)*sin(x-h/2)*cos(x-h/2) by
VALUED_1:5
    .= (1/4)*(-cos((x+h/2)+(x+h/2)-(x+h/2))+cos((x+h/2)+(x+h/2)-(x+h/2)) +
cos((x+h/2)+(x+h/2)-(x+h/2))-cos((x+h/2)+(x+h/2)+(x+h/2))) -sin(x-h/2)*sin(x-h/
  2)*cos(x-h/2) by SIN_COS4:34
    .= (1/4)*(cos(x+h/2)-cos(3*(x+h/2))) -(1/4)*(-cos((x-h/2)+(x-h/2)-(x-h/2
))+cos((x-h/2)+(x-h/2) -(x-h/2))+cos((x-h/2)+(x-h/2)-(x-h/2)) -cos((x-h/2)+(x-h
  /2)+(x-h/2))) by SIN_COS4:34
    .= (1/4)*(cos(x+h/2)-cos(x-h/2))-(1/4)*(cos(3*(x+h/2))-cos(3*(x-h/2)))
    .= (1/4)*(-2*(sin(((x+h/2)+(x-h/2))/2)*sin(((x+h/2)-(x-h/2))/2))) -(1/4)
  *(cos(3*(x+h/2))-cos(3*(x-h/2))) by SIN_COS4:18
    .= (1/4)*(-2*(sin(x)*sin(h/2))) -(1/4)*(-2*(sin(((y+z/2)+(y-z/2))/2) *
  sin(((y+z/2)-(y-z/2))/2))) by SIN_COS4:18
    .= -(1/2)*(sin(x)*sin(h/2))+(1/2)*(sin(3*x)*sin(3*h/2));
  hence thesis;
end;
