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 fD(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;
  fD(sin(#)cos(#)cos,h).x = (sin(#)cos(#)cos).(x+h) -(sin(#)cos(#)cos).x
  by DIFF_1:3
    .= ((sin(#)cos).(x+h))*(cos.(x+h)) -(sin(#)cos(#)cos).x by VALUED_1:5
    .= (sin.(x+h))*(cos.(x+h))*(cos.(x+h)) -(sin(#)cos(#)cos).x by VALUED_1:5
    .= (sin.(x+h))*(cos.(x+h))*(cos.(x+h)) -((sin(#)cos).x)*(cos.x) by
VALUED_1:5
    .= sin(x+h)*cos(x+h)*cos(x+h) -sin(x)*cos(x)*cos(x) by VALUED_1:5
    .= (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)))-sin(x)*cos(x)*cos(x) by SIN_COS4:35
    .= (1/4)*(sin(x+h)+sin(3*(x+h)))-(1/4) *(sin(x+x-x)-sin(x+x-x)+sin(x+x-x
  )+sin(x+x+x)) by SIN_COS4:35
    .= (1/4)*(sin(x+h)-sin(x))+(1/4)*(sin(3*(x+h))-sin(3*x))
    .= (1/4)*(2*(cos((x+h+x)/2)*sin((x+h-x)/2))) +(1/4)*(sin(3*(x+h))-sin(3*
  x)) by SIN_COS4:16
    .= (1/4)*(2*(cos((2*x+h)/2)*sin(h/2))) +(1/4)*(2*(cos((y+z+y)/2)*sin((y+
  z-y)/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;
