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 cD(f,h).x = 2*h*x
proof
  assume
A1:for x holds f.x = x^2;
  cD(f,h).x = f.(x+h/2) - f.(x-h/2) by DIFF_1:5
    .= (x+h/2)^2 - f.(x-h/2) by A1
    .= (x+h/2)^2 - (x-h/2)^2 by A1
    .= h*(2*x);
  hence thesis;
end;
