  Ahlfors' theory of covering surfaces is one of the major mathematical
achievement of last century. The most important part of his theory is the
Second Fundamental Theorem (SFT). We are interested in the relation of errors
of Ahlfors' SFT with the same boundary curve.
  In this paper we will prove a result which is used to establish the best
bound of the constant in Ahlfors' SFT in arXiv.2307.04623.
  Precisely speaking, we will prove that for any surface
$\Sigma\in\mathcal{F}_r(L,m)$, a new surface $\Sigma_1$ can be constructed
based on it, such that $R(\Sigma_1)\ge R(\Sigma)$ and $L(\partial\Sigma_1)\le
L(\partial\Sigma)$, where $R(\Sigma)$ is Ahlfors' error term and
$L(\partial\Sigma)$ is the boundary length of the surface $\Sigma$, and the
covering degree of $\Sigma_1$ has an upper bound independent of surfaces.
Meanwhile, this conclusion suggests that the supremum of
$H(\Sigma)=R(\Sigma)/L(\partial\Sigma)$ can be achieved by surfaces in the
space $\mathcal{F}_r'(L,m)$.
