Corollary of the Malgrange Preparation Theorem Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that
$$f(0,0)=0,\ \frac{\partial f}{\partial t} (0,0) = 0,\ldots, \frac{\partial^{k-1} f}{\partial t^{k-1}} (0,0) = 0,\ \frac{\partial^{k} f}{\partial t^{k}} (0,0) \neq 0,$$
then the Malgrange preparation theorem, states that there exists smooth functions $c,a_0,...,a_{k-1}:\mathbb{R^n}\to\mathbb{R}$, such that near the origin
$${\displaystyle f(t,x)=c(t,x)\left(t^{k}+a_{k-1}(x)t^{k-1}+\cdots +a_{0}(x)\right)},$$
and $c(0,0)\neq 0$.
I'm reading the paper "S. M Vishik - Vector Fields Near the Boundary of a Manifold", and on page $17$, the author says that the following theorem is a corollary of the Magrange preparation theorem.

Corollary: Let $\varphi:\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}$ a smooth function such that
  $$\varphi(0,0)=0,\ \frac{\partial \varphi}{\partial t} (0,0) = 0,\ldots, \frac{\partial^{k-1} \varphi}{\partial t^{k-1}} (0,0) = 0,\ \frac{\partial^{k} \varphi}{\partial t^{k}} (0,0) \neq 0,$$
  then there exists smooth functions $b,a_1,...,a_k:\mathbb{R}^n\times\mathbb{R} \to \mathbb{R}$ such that
  $$t^{k}+\sum_{i=1}^{k-1} a_i(x,\varphi(t,x))\cdot t^i = b(x,\varphi(t,x)), $$
  in a neighborhood of the origin.

Does anyone know this corollary and could you give me information on how to demonstrate it or let me know where I can find its demonstration?

Just some commentaries
This is the part that the author claims such corollary, $M$ is a manifold, $Q$ is a codimension 1 submanifold of $M$ and "shoots" is a fancy name for "germs".

And the numbered reference:

however, I was not able to find this paper.
 A: Too long for a comment.
I found mentioned Russian translation "Особенности дифференцируемых отображений" of Malgrange’s paper. Although Russian is my native language (so I can translate for you a few places of the paper, if needed) unfortunately, I am far from the topic of the paper and don’t understand it. 
I see the formula very similar (5) neither at p.  185 nor at other pages (183, 184,  186, 187, 188, 188).
Here is the mentioned corollary:

Here is its translation.
Let shoots [Sorry, I don’t know the right translation of this word. A main meaning of a russian word “ростки” is sprouts. AR] $\varphi_1,\dots,\varphi_p\in \mathcal E(x)$. Then the following conditions 
are  equivalent:
($a$)’ $\varphi_1,\dots,\varphi_p$ generate $\mathcal E(x)$ as a $\mathcal E(y)$-module.
($\hat a$)’ $\varphi_1,\dots,\hat\varphi_p$ generate $\hat{\mathcal E}(x)$ as a $\hat{\mathcal E}(y)$-module.
($b$)’ images $\varphi_1,\dots,\varphi_p$ in a ring  $\mathcal E(x)/ \mathcal E(x)u^*{\frak m}({\mathcal E}(y))$ generate it as a linear space over $\Bbb R$.
($\hat b$)’ images $\hat\varphi_1,\dots, \hat\varphi_p$ in a ring  $\hat{\mathcal E}(x)/\hat{\mathcal E}(x) \hat u^*{\frak m}(\hat{\mathcal E}(y))$ generate it as a linear space over $\Bbb R$.
Example 2. Weierstraß’ prepartion theorem. Let $F(x_1,\dots, x_n)\in\mathcal E_n$ be a regular shoot of order $p$  with respect to $x_n$ (that is $F(0,\dots, 0, x_n)$) has in a point $x_n=0$ zero of order exactly $p$). Moreover, let again $m=n$ and a shoot of a map $u$ is defined as 
$$(x_1,\dots, x_{n_1}, x_n)\to (x_1,\dots, x_{n-1}, F(x_1,\dots, x_n)).$$
Obviuously, an ideal generated in the ring $\hat{\mathcal E}_n$ by elements 
$x_1,\dots, x_{n-1}, \hat F$ coincides with an ideal generated in this ring by elements 
$x_1,\dots, x_{n-1}, x_n^p$. Now we can apply the corollary from theorem 1 (we need the equivalency of conditions ($a$)’ and ($\hat b$)’), putting $\varphi_i=x^{p-i}_n$ ($1\le i\le p$). In other words, each shoot $f\in\mathcal E_n$ can be written in the form
$$f(x_1,\dots, x_n)=\sum_{i=1}^p g_i(x_1,\dots, x_{n-1}, F)x^{p-i}_n,$$
where all $g_i\in\mathcal E_n$. Introduce a notation $h_i(x_1,\dots, x_{n-1})=g_i(x_1,\dots, x_{n-1},0)$ and remark that $g_i-h_i=x_nk_i$ for some $k_i\in\mathcal E_n$. Substituting $x_n$ by $F$, we obtain the following result: 
(W) Let $F\in\mathcal E_n$ be a regular shoot of order $p$ with respect to $x_n$. Than for any shoot $f\in\mathcal E_n$ there exist shoots $Q\in\mathcal E_n$ and $h_i\in\mathcal E_{n-1}$ such that 
$$f(x_1,\dots, x_n)=F(x_1,\dots, x_n)Q(x_1,\dots, x_n)+\sum_{i=1}^p h_i(x_1,\dots, x_{n-1})x^{p-i}_n.$$
Exactly the same claim for analytical functions constitutes a Weierstraß’ preparation theorem in Рюккерт’s form. The same theorem in Weierstraß’ form is obtained applying (W) to the case $f=x^p_n$;...
