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