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 = x^2) implies fD(f,h).x = 2*x*h + h^2
proof
  assume
A1:for x holds f.x = x^2;
then f.(x+h) = (x+h)^2;
  then fD(f,h).x = (x+h)^2 - f.x by DIFF_1:3
    .= x^2 + 2*x*h + h^2 - x^2 by A1
    .= 2*x*h + h^2;
  hence thesis;
end;
