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(#)sin(#)cos,x0,x1!] = -(1/2)*(sin(3*(x1+x0)/2)*sin(3*(x1-x0)/2)
  +sin((x0+x1)/2)*sin((x0-x1)/2))/(x0-x1)
proof
  set y = 3*x0;
  set z = 3*x1;
  [!sin(#)sin(#)cos,x0,x1!] = (((sin(#)sin).x0)*(cos.x0) -(sin(#)sin(#)cos
  ).x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0)*(sin.x0)*(cos.x0) -(sin(#)sin(#)cos).x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0)*(sin.x0)*(cos.x0) -((sin(#)sin).x1)*(cos.x1))/(x0-x1) by
VALUED_1:5
    .= (sin(x0)*sin(x0)*cos(x0) -sin(x1)*sin(x1)*cos(x1))/(x0-x1) by VALUED_1:5
    .= ((1/4)*(-cos(x0+x0-x0)+cos(x0+x0-x0) +cos(x0+x0-x0)-cos(x0+x0+x0)) -
  sin(x1)*sin(x1)*cos(x1))/(x0-x1) by SIN_COS4:34
    .= ((1/4)*(cos(x0)-cos(3*x0))-(1/4) *(-cos(x1+x1-x1)+cos(x1+x1-x1)+cos(
  x1+x1-x1) -cos(x1+x1+x1)))/(x0-x1) by SIN_COS4:34
    .= ((1/4)*(cos(x0)-cos(x1))+(1/4)*(cos(z)-cos(y)))/(x0-x1)
    .= ((1/4)*(-2*(sin((x0+x1)/2)*sin((x0-x1)/2))) +(1/4)*(cos(z)-cos(y)))/(
  x0-x1) by SIN_COS4:18
    .= ((1/4)*(-2*(sin((x0+x1)/2)*sin((x0-x1)/2)))+(1/4) *(-2*(sin((z+y)/2)*
  sin((z-y)/2))))/(x0-x1) by SIN_COS4:18
    .= (-(1/2)*(sin(3*(x1+x0)/2)*sin(3*(x1-x0)/2) +sin((x0+x1)/2)*sin((x0-x1
  )/2)))/(x0-x1);
  hence thesis by XCMPLX_1:187;
end;
