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
  [!sin(#)cos,x0,x1!] = (1/2)*(sin(2*x0)-sin(2*x1))/(x0-x1)
proof
  [!sin(#)cos,x0,x1!] = ((sin.x0)*(cos.x0)-(sin(#)cos).x1) /(x0-x1) by
VALUED_1:5
    .= (sin(x0)*cos(x0)-sin(x1)*cos(x1))/(x0-x1) by VALUED_1:5
    .= ((1/2)*(sin(x0+x0)+sin(x0-x0)) -sin(x1)*cos(x1))/(x0-x1) by SIN_COS4:30
    .= ((1/2)*(sin(x0+x0)+sin(x0-x0)) -(1/2)*(sin(x1+x1)+sin(x1-x1)))/(x0-x1
  ) by SIN_COS4:30
    .= (1/2)*(sin(2*x0)-sin(2*x1))/(x0-x1);
  hence thesis;
end;
