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 bD(sin(#)sin,h).x = (1/2)*(cos(2*(x-h))-cos(2*x))
proof
  let x;
  bD(sin(#)sin,h).x = (sin(#)sin).x-(sin(#)sin).(x-h) by DIFF_1:4
    .= (sin.x)*(sin.x)-(sin(#)sin).(x-h) by VALUED_1:5
    .= sin(x)*sin(x)-sin(x-h)*sin(x-h) by VALUED_1:5
    .= -(1/2)*(cos(x+x)-cos(x-x))-sin(x-h)*sin(x-h) by SIN_COS4:29
    .= -(1/2)*(cos(2*x)-cos(0))-(-(1/2) *(cos((x-h)+(x-h))-cos((x-h)-(x-h)))
  ) by SIN_COS4:29;
  hence thesis;
end;
