\label{eq:xN-1 for DN}
[x^{N-1}]=\frac{v_{N+1}}{2}[w]+\frac{v_N}{2}[y]-\sum_{k=1}^{N-1}(k-1)v_k[x^{k-1}],\end{gathered}$$ that allows to express any element $[x^p]$ with $p\ge N-1$ in terms of the elements $[1],[x],\ldots,[x^{N-2}]$, $[y]$, $[w]$.

Identifying the ${{\mathbb{C}}}[v_1,\dots,v_N,v_{N+1},v_{N+1}^{-1}]$-modules ${\mathcal{T}}^{\mathrm{alg}}_{M^{\mathrm{ext}}_{D_N}}(M^{\mathrm{ext}}_{D_N})$ and ${{\widehat{\mathcal{A}}}}^{\mathrm{ext}}_{D_N}$ via the isomorphism $\Psi^{\mathrm{ext}}_{D_N}$ defined by $$\begin{gathered}
\Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_k}\right) := [x^{k-1}], \quad 1 \le k \le N-1,\qquad \Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_N} \right) := [y],\qquad \Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_{N+1}} \right) := [w],\end{gathered}$$ we endow the tangent spaces $T_pM_{D_N}^{\mathrm{ext}}$ with a multiplication and, clearly, the structure constants of it belong to the ring ${\mathbb C}[v_1,\ldots,v_N,v_{N+1},v_{N+1}^{-1}]$.

Consider the flat coordinates $t^\alpha=t^\alpha(v_1,\dots,v_N)$, $1 \le \alpha \le N$, and the potential $F_{D_N}(t^1,\ldots,t^N)$ of the Frobenius manifold of the singularity $D_N$. Let $t^{N+1}:=v_{N+1}$ and define a function $F^o_{D_N}(t^1,\ldots,t^{N+1})$ by $$\begin{gathered}
F^o_{D_N}:=\left.\left(\sum_{k=1}^{N-1} \frac{v_k v_{N+1}^{2 k-1}}{2^{k-1}(2 k-1)} + \frac{v_{N+1}^{2 N-1}}{2^{N-2}(2 N-1) (2N-2)} + \frac{v_N^2}{2 v_{N+1}}\right)\right|_{v_i=v_i(t^*)}.\end{gathered}$$

 \
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 $M^{\mathrm{ext}}_{D_N}$, constructed above, define a flat F-manifold structure on $M^{\mathrm{ext}}_{D_N}$.\
2. The vector potential of this flat F-manifold is given by $\left(\eta_{D_N}^{1\alpha} \frac{\d F_{D_N}}{\d t^\alpha}, \ldots, \eta_{D_N}^{N\alpha} \frac{\d F_{D_N}}{\d t^\alpha}, F^o_{D_N}\right)$.

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: 