Graduate Help in understanding this derivation of Lagrange Equations in Non-Holonomic case

[Kashmir]

I dont Understand how we get the final equations relating $Q_r$ with $\lambda$ given the conditions above?

 

[wrobel]

There is a nice theorem.
Let $f,f_1,\ldots, f_n:X\to\mathbb{R}$ be linear functionals defined on a vector space $X$.
Theorem. Assume that $$\bigcap_{k\in\{1,\ldots,n\}}\ker f_k\subset \ker f.$$
Then there are constants $\lambda_1,\ldots,\lambda_n$ such that
$$f=\sum_{k=1}^n\lambda_k f_k.$$
Moreover this theorem is a special case of the following fact. Let $X,Y,Z$ be vector spaces perhaps infinite dimensional. Let
$$A:X\to Y,\quad B:X\to Z$$ be linear operators such that $\ker A\subset\ker B$. Then there is a linear operator $\Lambda:Y\to Z$ such that $B=\Lambda A$.