How is the implicit function theorem used here. I am reading a paper in which it says the following:

Here, $\Gamma$ is an analytic a curve in $\mathbb{C}^n$ defined near the origin. Write $z = x +iy$ and suppose that for each pair $(t_x, t_y) \in T_0(\Gamma)$ were have $|t_x|^2 \geq |t_y|^2$ (here, $T_0$ denotes the Tangent space). The author claims that by the implicit function theorem there exists a neighbourhood of the origin where $\Gamma$ is given by
  $$
\{x+itH(x) : x\in W\}
$$
  where $W\subseteq \mathbb{R}^n$ and $H$ is real analytic.

I am not sure how this follows from the implicit function theorem. I'd appreciate it if anyone could clarify what is going on.
Thanks in advance!

The map $z=x+iy \mapsto x$ is a submersion, and its differential has kernel $\{iy|y \in \mathbb{R}^n\}$. So the map is a local diffeomorphism on $\Gamma$ near $0$, as the kernel of the differential is zero, so the differential is an isomorphism of real vector spaces. It follows that there is a local inverse $x \mapsto (x,y(x))$.
