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
  fD(sin(#)sin(#)sin,h).x
  =(1/2)*(3*cos((2*x+h)/2)*sin(h/2)-cos(3*(2*x+h)/2)*sin(3*h/2))
proof
  fD(sin(#)sin(#)sin,h).x = (sin(#)sin(#)sin).(x+h)
       -(sin(#)sin(#)sin).x by DIFF_1:3
    .= ((sin(#)sin).(x+h))*(sin.(x+h))
       -(sin(#)sin(#)sin).x by VALUED_1:5
    .= (sin.(x+h))*(sin.(x+h))*(sin.(x+h))
       -(sin(#)sin(#)sin).x by VALUED_1:5
    .= (sin.(x+h))*(sin.(x+h))*(sin.(x+h))
       -((sin(#)sin).x)*(sin.x) by VALUED_1:5
    .= sin(x+h)*sin(x+h)*sin(x+h)
       -sin(x)*sin(x)*sin(x) by VALUED_1:5
    .= (1/4)*(sin((x+h)+(x+h)-(x+h))+sin((x+h)+(x+h)-(x+h))
       +sin((x+h)+(x+h)-(x+h))-sin((x+h)+(x+h)+(x+h)))
       -sin(x)*sin(x)*sin(x) by SIN_COS4:33
    .= (1/4)*(sin(x+h)+sin(x+h)+sin(x+h)-sin(3*(x+h)))
       -(1/4)*(sin(x+x-x)+sin(x+x-x)+sin(x+x-x)-sin(x+x+x)) by SIN_COS4:33
    .= (1/4)*(3*(sin(x+h)-sin(x))-(sin(3*(x+h))-sin(3*x)))
    .= (1/4)*(3*(2*(cos((x+h+x)/2)*sin((x+h-x)/2)))
       -(sin(3*(x+h))-sin(3*x))) by SIN_COS4:16
    .= (1/4)*(3*(2*(cos((x+h+x)/2)*sin((x+h-x)/2)))
       -(2*(cos((3*(x+h)+3*x)/2)*sin((3*(x+h)-3*x)/2)))) by SIN_COS4:16
    .= (1/2)*(3*cos((2*x+h)/2)*sin(h/2)-cos(3*(2*x+h)/2)*sin(3*h/2));
  hence thesis;
end;
