$F:R^3 \to R^3 $ and $F \in C^1$ and let $a$ be a point where $F(a)=0$ and
$ DF|a = \begin{bmatrix} \partial f_1/\partial x & \partial f_1/\partial y & \partial f_1/\partial z \\ \partial f_2/\partial x & \partial f_2/\partial y & \partial f_2/\partial z \\ \partial f_3/\partial x & \partial f_3/\partial y & \partial f_3/\partial z \\ \end{bmatrix} $
where rank($DF|a)=2 \\$ let us assume that the determinant of $2\times2$ of the top right corner is non zero by applying the implicit function theorem there is a function $\varphi$ defined in D and $a\in D$ where
$f_1(x,\varphi(x))=0$ and $f_2(x,\varphi(x))=0$
I try to prove that there is $D2 \subseteq D$ where $f_3(x,\varphi(x))=0$
my intuition is that $F$ locally in point $a$ is similar to linear function to $DF|a$ and because the rank is 2 and $F(0)=0$ it is true that
$f_3=\alpha f_1 + \beta f_2$ at the point a
My intuition tells me it should be true for $D2.$