reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  cD(cos(#)cos(#)cos,h).x = -(1/2)*(3*sin(x)*sin(h/2)+sin(3*x)*sin(3*h/2))
proof
  cD(cos(#)cos(#)cos,h).x = (cos(#)cos(#)cos).(x+h/2)
       - (cos(#)cos(#)cos).(x-h/2) by DIFF_1:5
    .= ((cos(#)cos).(x+h/2))*(cos.(x+h/2))
       -(cos(#)cos(#)cos).(x-h/2) by VALUED_1:5
    .= (cos.(x+h/2))*(cos.(x+h/2))*(cos.(x+h/2))
       -(cos(#)cos(#)cos).(x-h/2) by VALUED_1:5
    .= (cos.(x+h/2))*(cos.(x+h/2))*(cos.(x+h/2))
       -((cos(#)cos).(x-h/2))*(cos.(x-h/2)) by VALUED_1:5
    .= cos(x+h/2)*cos(x+h/2)*cos(x+h/2)
       -cos(x-h/2)*cos(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)))
       -cos(x-h/2)*cos(x-h/2)*cos(x-h/2) by SIN_COS4:36
    .= (1/4)*(cos(x+h/2)+cos(x+h/2)+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_COS4:36
    .= (1/4)*(3*(cos(x+h/2)-cos(x-h/2))+(cos(3*(x+h/2))-cos(3*(x-h/2))))
    .= (1/4)*(3*(-2*(sin(((x+h/2)+(x-h/2))/2)*sin(((x+h/2)-(x-h/2))/2)))
       +(cos(3*(x+h/2))-cos(3*(x-h/2)))) by SIN_COS4:18
    .= (1/4)*(3*(-2*(sin(2*x/2)*sin(h/2)))
       +(-2*(sin(((3*(x+h/2))+(3*(x-h/2)))/2)
       *sin(((3*(x+h/2))-(3*(x-h/2)))/2)))) by SIN_COS4:18
    .= -((1/2))*(3*(sin(x)*sin(h/2))+(sin(3*x)*sin(3*h/2)));
  hence thesis;
end;
