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<>0 & x-h<>0
  implies bD(f,h).x = (-k)*h*(2*x-h)/((x^2-x*h)^2)
proof
  assume that
A1:for x holds f.x = k/(x^2) and
A2:x<>0 & x-h<>0;
A3:f.(x-h) = k/((x-h)^2) by A1;
  bD(f,h).x = f.x - f.(x-h) by DIFF_1:4
    .= k/(x^2) - k/((x-h)^2) by A1,A3
    .= (k*((x-h)^2)-k*(x^2))/((x^2)*((x-h)^2)) by A2,XCMPLX_1:130
    .= (-k)*h*(2*x-h)/((x^2-x*h)^2);
  hence thesis;
end;
