Proof of lemma 1 Paillier encryption In the original paper of Paillier, lemma 1 shows why $n$ must divide the order of $g$. What I don't understand in the proof of this lemma is why $g^{x_2-x_1}(y_2/y_1)^n$ implies $g^{\lambda(x_2-x_1)}$. Where does this result come from?
This is obtained by raising to $\lambda=\lambda(n)$: since the order of any element in $\mathbb{Z}_{n^2}$  divides $n\cdot\lambda$, the second part cancels out:

$$\begin{align}
    g^{x_1-x_2}\cdot(y_2/y_1)^n=1\bmod{n^2} &amp;\Leftrightarrow\\
   g^{(x_1-x_2)\cdot\lambda}\cdot(y_2/y_1)^{n\cdot\lambda}=1\bmod{n^2}&amp;\Leftrightarrow\\
  g^{(x_1-x_2)\cdot\lambda}\cdot 1=1\bmod{n^2}&amp;\Leftrightarrow\\
g^{(x_1-x_2)\cdot\lambda}=1\bmod{n^2}
   \end{align}$$