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(#)cos,h).x = (1/2)*(sin(2*x)-sin(2*(x-h)))
proof
  let x;
  bD(sin(#)cos,h).x = (sin(#)cos).x-(sin(#)cos).(x-h) by DIFF_1:4
    .= (sin.x)*(cos.x)-(sin(#)cos).(x-h) by VALUED_1:5
    .= sin(x)*cos(x)-sin(x-h)*cos(x-h) by VALUED_1:5
    .= (1/2)*(sin(x+x)+sin(x-x))-sin(x-h)*cos(x-h) by SIN_COS4:30
    .= (1/2)*(sin(x+x)+sin(0)) -(1/2)*(sin((x-h)+(x-h))+sin((x-h)-(x-h))) by
SIN_COS4:30
    .= (1/2)*sin(2*x)-(1/2)*sin(2*(x-h));
  hence thesis;
end;
