proof.

Since algebra ${\tilde{\mathfrak{g}}}_1$ is commutative, for $\gamma, \delta \in \Gamma$ vertex operator $Y(x_{\gamma}(-1)x_{\delta}(-1)v_{\Lambda_0},z)$ equals to the product of $x_{\gamma}(z)$ and $x_{\delta}(z)$ as ordinary Laurent series (cf. [@DL]). In this way we get the relations between vertex operators on level 1 modules.

Let $\gamma, \delta \in \Gamma$. We have: $$\begin{aligned}
\label{relacije}
& & \label{rel1} x_{\delta}(z)x_{\gamma}(z)  = 0 \qquad \textrm{if} \quad \delta \neq \underline{\gamma}, \\
& & \label{rel2} x_{\underline{\gamma}}(z) x_{\gamma}(z)  = C \cdot x_{\underline{\delta}}(z) x_{\delta}(z) \qquad \textrm{and}\\
& & \label{rel3} x_{\tau}(z) x_{\underline{\gamma}}(z) x_{\gamma}(z)  = 0.\end{aligned}$$

Difference and initial conditions for level 1 modules {#pocetak1}
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Now, using relations between fields from the previous section, we reduce the spanning set of the Feigin-Stoyanovsky’s type subspaces of level 1 modules, i.e. $L(\Lambda_0)$, $L(\Lambda_1)$, $L(\Lambda_{{\Lambda_{\ell-1}}})$ and $L({\Lambda_{\ell}})$. We further observe which factors of $-1$ degree, i.e. $x_{\gamma}(-1)$ factors, annihilate the highest weight vectors or can be expressed as a sum of greater monomials on this vectors.

Difference conditions
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The coefficients of $z^{n-2}$ in the equalities and give us the relations between the infinite sums of monomials from $U({\tilde{\mathfrak{g}}}_1)$: $$\begin{aligned}
0 &= x_{\delta}(z)x_{\gamma}(z) = \sum_n \left( \sum_{p+s=n}  x_{\delta}(-p)x_{\gamma}(-s) \right) z^{n-2} = \label{rell1} \\ &= \sum_n \left( \sum_p x_{\delta}(-n+p) x_{\gamma}(-p) \right) z^{n-2} \notag \end{aligned}$$ and 