multiplication and, clearly, the structure constants of it are polynomials in the coordinates $v_1,\ldots,v_{N+1}$.

Consider the flat coordinates $t^\alpha=t^\alpha(v_1,\dots,v_N)$, $1 \le \alpha \le N$, the potential $F_{A_N}(t^1,\ldots,t^N)$ of the Frobenius manifold of the singularity $A_N$ and the function $F^o_{A_N}$, described in Section \[section:extended r-spin\].

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1. The coordinates $t^1(v_1,\ldots,v_N),\ldots,t^N(v_1,\ldots,v_N)$ and $t^{N+1}:=v_{N+1}$ together with the multiplicative structure on $M^{\mathrm{ext}}_{A_N}$, constructed above, define a flat F-manifold structure on $M^{\mathrm{ext}}_{A_N}$.\
2. The vector potential of this flat F-manifold is given by $\left(\eta_{A_N}^{1\alpha} \frac{\d F_{A_N}}{\d t^\alpha}, \ldots, \eta_{A_N}^{N\alpha} \frac{\d F_{A_N}}{\d t^\alpha}, F^o_{A_N}\right)$, where we identify $s=t^{N+1}$.

We denote by $(c^{\mathrm{ext}})^\alpha_{\beta\gamma}$ the structure constants of multiplication in the coordinates $t^1,\ldots,t^{N+1}$ and by $(c^{\mathrm{ext}}_v)^\alpha_{\beta\gamma}$ the structure constants of multiplication in the coordinates $v_1,\ldots,v_{N+1}$.

In order to prove the theorem, we have to check the following equations: $$\begin{aligned}
(c^{\mathrm{ext}})^\alpha_{\beta\gamma}=&\sum_{\mu=1}^N\eta_{A_N}^{\alpha\mu} \frac{\d^3 F_{A_N}}{\d t^\mu\d t^\beta\d t^\gamma},&& 1\le\alpha\le N,&& 1\le\beta,\gamma\le N+1,\label{eq:first property for AN}\\
(c^{\mathrm{ext}})^{N+1}_{\alpha\beta}=&\frac{\d^2 F^o_{A_N}}{\d t^\alpha\d t^\beta},&& 1\le\alpha,\beta\le N+1.&&\label{eq:second property for AN}\end{aligned}$$ Since the subspace ${\mathbb C}[v_1,\ldots,v_{N+1}]{\left<}[w]{\right>}$ is an ideal in the ring ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}$ and the quotient by this ideal coincides with the ring ${{\widehat{\mathcal{A}}}}_{A_N}$, we have $$(c^{\mathrm{ext}}_v)^a_{b,c}=
\begin{cases}
(c_v)^a_{b,c},&\text{if $1\le a,b,c\le N$},\\
0,&\text{if $1\le a\le N$ and one of the indices $b,c$ is equal to $N+1$}.
\end{cases}$$ This 