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(cos(#)cos,h).x = (1/2)*(cos(2*(x+h))-cos(2*x))
proof
  let x;
  fD(cos(#)cos,h).x = (cos(#)cos).(x+h) -(cos(#)cos).x by DIFF_1:3
    .= (cos.(x+h))*(cos.(x+h))-(cos(#)cos).x by VALUED_1:5
    .= cos(x+h)*cos(x+h)-cos(x)*cos(x) by VALUED_1:5
    .= (1/2)*(cos((x+h)+(x+h))+cos((x+h)-(x+h))) -cos(x)*cos(x) by SIN_COS4:32
    .= (1/2)*(cos(2*(x+h))+cos(0)) -(1/2)*(cos(x+x)+cos(x-x)) by SIN_COS4:32
    .= (1/2)*cos(2*(x+h))-(1/2)*cos(2*x);
  hence thesis;
end;
