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 fD(f,h).x = (-k)*h*(2*x+h)/((x^2+h*x)^2)
proof
  assume that
A1:for x holds f.x = k/(x^2) and
A2:x<>0 & x+h<>0;
  fD(f,h).x = f.(x+h) - f.x by DIFF_1:3
    .= k/((x+h)^2) - f.x by A1
    .= k/((x+h)^2) - k/(x^2) by A1
    .= (k*(x^2) - k*((x+h)^2))/(((x+h)^2)*(x^2)) by A2,XCMPLX_1:130
    .= (-k)*h*(2*x+h)/((x^2+h*x)^2);
  hence thesis;
end;
