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(#)cos,x0,x1!] = (1/2)*(cos((x0+x1)/2)*sin((x0-x1)/2) +cos(
  3*(x0+x1)/2)*sin(3*(x0-x1)/2))/(x0-x1)
proof
  set y = 3*x0;
  set z = 3*x1;
  [!sin(#)cos(#)cos,x0,x1!] = (((sin(#)cos).x0)*(cos.x0) -(sin(#)cos(#)cos
  ).x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0)*(cos.x0)*(cos.x0) -(sin(#)cos(#)cos).x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0)*(cos.x0)*(cos.x0) -((sin(#)cos).x1)*(cos.x1))/(x0-x1) by
VALUED_1:5
    .= (sin(x0)*cos(x0)*cos(x0) -sin(x1)*cos(x1)*cos(x1))/(x0-x1) by VALUED_1:5
    .= ((1/4)*(sin(x0+x0-x0)-sin(x0+x0-x0)+sin(x0+x0-x0) +sin(x0+x0+x0))-sin
  (x1)*cos(x1)*cos(x1))/(x0-x1) by SIN_COS4:35
    .= ((1/4)*(sin(x0)+sin(3*x0)) -(1/4)*(sin(x1+x1-x1)-sin(x1+x1-x1) +sin(
  x1+x1-x1)+sin(x1+x1+x1)))/(x0-x1) by SIN_COS4:35
    .= ((1/4)*(sin(x0)-sin(x1))+(1/4)*(sin(3*x0)-sin(3*x1)))/(x0-x1)
    .= ((1/4)*(2*(cos((x0+x1)/2)*sin((x0-x1)/2))) +(1/4)*(sin(y)-sin(z)))/(
  x0-x1) by SIN_COS4:16
    .= ((1/2)*(cos((x0+x1)/2)*sin((x0-x1)/2))+(1/4) *(2*(cos((y+z)/2)*sin((y
  -z)/2))))/(x0-x1) by SIN_COS4:16
    .= (1/2)*(cos((x0+x1)/2)*sin((x0-x1)/2) +cos(3*(x0+x1)/2)*sin(3*(x0-x1)/
  2))/(x0-x1);
  hence thesis;
end;
