An alternative proof of ${\widehat{\mathfrak{s}\mathfrak{l}}}_2^{\mathrm{\prime }}$ standard module semi-infinite structure

B. Feigin and A. Stoyanovsky found the basis of semi-infinite monomials in standard ${\widehat{\mathfrak{s}\mathfrak{l}}}_2^{\mathrm{\prime }}$-module $L_{(0,1)}$ with Lefschetz formula for the corresponding flag variety. These semi-infinite monomials are constructed by modes of the current $e(z)=\sum _{n\in \mathbb{Z}}{e}_n{z}^{-n-1}$. We give an alternative proof of this fact using explicit fermionic construction of this module. Namely, we realize $L_{(0,1)}$ inside of the zero-charge subspace of Fermionic Fock space and show linear independence of vectors corresponding to semi-infinite monomials.