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 bD(f,h).x = h*(2*x-h)
proof
  assume
A1:for x holds f.x = x^2;
then A2:f.(x-h) = (x-h)^2;
  bD(f,h).x = f.x - f.(x-h) by DIFF_1:4
    .= x^2 - (x-h)^2 by A1,A2
    .= h*(2*x - h);
  hence thesis;
end;
