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
  (for x holds f.x=(tan(#)tan).x) & x+h/2 in dom tan & x-h/2 in dom tan implies
  cD(f,h).x = -(1/2)*(cos(h+2*x)-cos(h-2*x))/((cos(x+h/2)*cos(x-h/2))^2)
proof
  assume that
A1:for x holds f.x=(tan(#)tan).x and
A2:x+h/2 in dom tan & x-h/2 in dom tan;
A3:cos(x+h/2)<>0 & cos(x-h/2)<>0 by A2,FDIFF_8:1;
  cD(f,h).x = f.(x+h/2) - f.(x-h/2) by DIFF_1:5
    .= (tan(#)tan).(x+h/2) - f.(x-h/2) by A1
    .= (tan(#)tan).(x+h/2) - (tan(#)tan).(x-h/2) by A1
    .= tan.(x+h/2)*tan.(x+h/2)-(tan(#)tan).(x-h/2) by VALUED_1:5
    .= tan.(x+h/2)*tan.(x+h/2)-tan.(x-h/2)*tan.(x-h/2) by VALUED_1:5
    .= sin.(x+h/2)*(cos.(x+h/2))"*tan.(x+h/2)-tan.(x-h/2)*tan.(x-h/2)
                                                      by A2,RFUNCT_1:def 1
    .= sin.(x+h/2)*(cos.(x+h/2))"*(sin.(x+h/2)*(cos.(x+h/2))")
       -tan.(x-h/2)*tan.(x-h/2) by A2,RFUNCT_1:def 1
    .= sin.(x+h/2)*(cos.(x+h/2))"*(sin.(x+h/2)*(cos.(x+h/2))")
       -(sin.(x-h/2)*(cos.(x-h/2))")*tan.(x-h/2) by A2,RFUNCT_1:def 1
    .= (tan(x+h/2))^2-(tan(x-h/2))^2 by A2,RFUNCT_1:def 1
    .= (tan(x+h/2)-tan(x-h/2))*(tan(x+h/2)+tan(x-h/2))
    .= (sin((x+h/2)-(x-h/2))/(cos(x+h/2)*cos(x-h/2)))*(tan(x+h/2)+tan(x-h/2))
                                                            by A3,SIN_COS4:20
    .= (sin(h)/(cos(x+h/2)*cos(x-h/2)))
       *(sin((x+h/2)+(x-h/2))/(cos(x+h/2)*cos(x-h/2))) by A3,SIN_COS4:19
    .= (sin(h)*sin(2*x))/((cos(x+h/2)*cos(x-h/2))^2) by XCMPLX_1:76
    .= (-(1/2)*(cos(h+(2*x))-cos(h-(2*x))))/((cos(x+h/2)*cos(x-h/2))^2)
                                                          by SIN_COS4:29
    .= -(1/2)*(cos(h+2*x)-cos(h-2*x))/ ((cos(x+h/2)*cos(x-h/2))^2);
  hence thesis;
end;
