reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  [!sin(#)sin(#)sin,x0,x1!] = (1/2)*(3*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(#)sin(#)sin,x0,x1!] = (((sin(#)sin).x0)*(sin.x0)
       -(sin(#)sin(#)sin).x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0)*(sin.x0)*(sin.x0)
       -(sin(#)sin(#)sin).x1)/(x0-x1) by VALUED_1:5
    .= ((sin.x0)*(sin.x0)*(sin.x0)
       -((sin(#)sin).x1)*(sin.x1))/(x0-x1) by VALUED_1:5
    .= (sin(x0)*sin(x0)*sin(x0)
       -sin(x1)*sin(x1)*sin(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)*sin(x1)*sin(x1))/(x0-x1) by SIN_COS4:33
    .= ((1/4)*(3*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_COS4:33
    .= ((1/4)*(3*(sin(x0)-sin(x1))-(sin(y)-sin(z))))/(x0-x1)
    .= ((1/4)*(3*(2*(cos((x0+x1)/2)*sin((x0-x1)/2)))
       -(sin(y)-sin(z))))/(x0-x1) by SIN_COS4:16
    .= ((1/4)*(3*(2*(cos((x0+x1)/2)*sin((x0-x1)/2)))
       -(2*(cos((y+z)/2)*sin((y-z)/2)))))/(x0-x1) by SIN_COS4:16
    .= (1/2)*(3*cos((x0+x1)/2)*sin((x0-x1)/2)
       -cos(3*(x0+x1)/2)*sin(3*(x0-x1)/2))/(x0-x1);
  hence thesis;
end;
