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 fD(sin(#)sin,h).x = (1/2)*(cos(2*x)-cos(2*(x+h)))
proof
  let x;
  fD(sin(#)sin,h).x = (sin(#)sin).(x+h)-(sin(#)sin).x by DIFF_1:3
    .= (sin.(x+h))*(sin.(x+h))-(sin(#)sin).x by VALUED_1:5
    .= sin(x+h)*sin(x+h)-sin(x)*sin(x) by VALUED_1:5
    .= -(1/2)*(cos((x+h)+(x+h))-cos((x+h)-(x+h))) -sin(x)*sin(x) by SIN_COS4:29
    .= -(1/2)*(cos(2*(x+h))-cos(0)) -(-(1/2)*(cos(x+x)-cos(x-x))) by
SIN_COS4:29;
  hence thesis;
end;
