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 = k/(x^2)) & x+h/2<>0 & x-h/2<>0
  implies cD(f,h).x = (-2*h*k*x)/((x^2-(h/2)^2)^2)
proof
  assume that
A1:for x holds f.x = k/(x^2) and
A2:x+h/2<>0 & x-h/2<>0;
  cD(f,h).x = f.(x+h/2) - f.(x-h/2) by DIFF_1:5
    .= k/((x+h/2)^2) - f.(x-h/2) by A1
    .= k/((x+h/2)^2) - k/((x-h/2)^2) by A1
    .= (k*((x-h/2)^2)-k*((x+h/2)^2))/(((x+h/2)^2)*((x-h/2)^2))
                                             by A2,XCMPLX_1:130
    .= (-2*h*k*x)/((x^2-(h/2)^2)^2);
  hence thesis;
end;
