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
  [!cos(#)cos,x0,x1!] = (1/2)*(cos(2*x0)-cos(2*x1))/(x0-x1)
proof
  [!cos(#)cos,x0,x1!] = ((cos.x0)*(cos.x0) -(cos(#)cos).x1)/(x0-x1) by
VALUED_1:5
    .= (cos(x0)*cos(x0) -cos(x1)*cos(x1))/(x0-x1) by VALUED_1:5
    .= ((1/2)*(cos(x0+x0)+cos(x0-x0)) -cos(x1)*cos(x1))/(x0-x1) by SIN_COS4:32
    .= ((1/2)*(cos(x0+x0)+cos(x0-x0)) -(1/2)*(cos(x1+x1)+cos(x1-x1)))/(x0-x1
  ) by SIN_COS4:32
    .= ((1/2)*cos(2*x0)-(1/2)*cos(2*x1))/(x0-x1);
  hence thesis;
end;
