Mass transfert in process chemistry engineering, two film theory
You already derived this relation (eq 1),
\begin{align} (y_1-y_1^*)&amp;=(y_1-y_1^I)+(y_1^I-y_1^*)\\ &amp;=(y_1-y_1^I)+ K_1^{eq}\left(x_1^I-x_1\right)\end{align}

Now, to keep the molar flux density between the gas phase concentration $y_1$ and fictitious conc. on the other side $y_1^*$ we assumed a fictitious overall transfer coefficient $K_{cg}$. So, now we have another relation for molar flux density
$$N_{1n}=K_{cg}c_g\times(y_1-y_1^*)$$
that follows,
$$y_1-y_1^*=\frac{N_{1n}}{K_{cg}c_g}$$
Similarly,
$$y_1-y_1^I=\frac{N_{1n}}{k_{cI}c_I}$$ 
and
$$x_1^I-x_1=\frac{N_{1n}}{k_{cg}c_g}$$ 
Plug these above three relations into eq 1 and you will get your last equation.