  The objective of this study is to correlate the scaling factor of the
Standard Model (SM) like Higgs boson and the cross section ratio of the process
$e^+ e^- \rightarrow hhf\overline{f}$ where $f\neq t$, normalized to SM
predictions in the type I of the Two Higgs Doublet Model. All calculations have
been performed at $\sqrt{s}=500$ GeV and $1 \leq \tan{\beta} \leq 30$ for
masses $m_H = m_A = m_{H^\pm} = 300GeV$ and $m_H=300$ GeV, $m_A=m_{H^\pm}=500$
GeV. The working scenario is by taking without alignment limit, that is
$s_{\beta-\alpha}= 0.98$ and $s_{\beta-\alpha}= 0.99,$ $0.995$, which gives the
enhancement in the cross section, particularly a few times greater than the
predictions of the SM due to resonant-impacts of the additional heavy neutral
Higgs bosons. This shows that enhancement in cross section occurs on leaving
the alignment i.e., $s_{\beta-\alpha}=1$, at which all the higgs that couple to
vector bosons and fermions have the same values as in SM at tree level. A large
value of enhancement factor is obtained at $s_{\beta-\alpha}= 0.98$ compared to
$s_{\beta-\alpha}= 0.99,$ $0 .995$. Furthermore, the decrease in the
enhancement factor is observed for the case when $m_H=300$ GeV,
$m_A=m_{H^\pm}=500$ GeV. The behavior of the scaling factor with $\tan{\beta}$
is also studied, which shows that for large values of $\tan\beta$, the scaling
factor becomes equal to $s_{\beta-\alpha}$. Finally a convincing correlation is
achieved by taking into account, the experimental and theoretical constraints
e.g, perturbative unitarity, vacuum stability and electroweak oblique
parameters.
