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(sin(#)sin(#)sin,h).x
  = (1/2)*(3*cos((2*x-h)/2)*sin(h/2)-cos(3*(2*x-h)/2)*sin(3*h/2))
proof
  bD(sin(#)sin(#)sin,h).x = (sin(#)sin(#)sin).x
       -(sin(#)sin(#)sin).(x-h) by DIFF_1:4
    .= ((sin(#)sin).x)*(sin.x)-(sin(#)sin(#)sin).(x-h) by VALUED_1:5
    .= (sin.x)*(sin.x)*(sin.x)-(sin(#)sin(#)sin).(x-h) by VALUED_1:5
    .= (sin.x)*(sin.x)*(sin.x)
       -((sin(#)sin).(x-h))*(sin.(x-h)) by VALUED_1:5
    .= sin(x)*sin(x)*sin(x)-sin(x-h)*sin(x-h)*sin(x-h) by VALUED_1:5
    .= (1/4)*(sin(x+x-x)+sin(x+x-x)+sin(x+x-x)
       -sin(x+x+x))-sin(x-h)*sin(x-h)*sin(x-h) by SIN_COS4:33
    .= (1/4)*(sin(x)+sin(x)+sin(x)-sin(3*x))
       -(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))) by SIN_COS4:33
    .= (1/4)*(3*(sin(x)-sin(x-h))-(sin(3*x)-sin(3*(x-h))))
    .= (1/4)*(3*(2*(cos((x+(x-h))/2)*sin((x-(x-h))/2)))
       -(sin(3*x)-sin(3*(x-h)))) by SIN_COS4:16
    .= (1/4)*(3*(2*(cos((2*x-h)/2)*sin(h/2)))
       -(2*(cos((3*x+3*(x-h))/2)*sin((3*x-3*(x-h))/2)))) by SIN_COS4:16
    .= (1/2)*(3*cos((2*x-h)/2)*sin(h/2)-cos(3*(2*x-h)/2)*sin(3*h/2));
  hence thesis;
end;
