Question in proof of open mapping theorem

Question

Why is the mapping $\tilde A: X/N \to Y,\ \tilde A(x+N)=A(x)$ one to one (injective) and onto (surjective)?

$A$ is a continuous, linear map from a F-space $X$ to a topological vector space $Y$ and $A(X)$ is second category in $Y$.

Clearly it is surjective because we showed that $A(X)=Y$ and therefore $A$ is surjective. But I don't understand why $\tilde A$ is injective.

Answer 1

Because\begin{align}\tilde A(x+N)=\tilde A(y+N)&\iff A(x)=A(y)\\&\iff A(x-y)=0\\&\iff x-y\in N\\&\iff x+N=y+N.\end{align}

Answer 2

$\tilde {A} (x+N)=\tilde {A} (y+N)$ implies that $Ax=Ay$, so $x-y \in N$. This implies that $x+N=y+N$.