I'm trying to solve this step from a Bourbaki's commutative algebra exercise (Chapter I, Flat Modules, exercise 23 (a) page 47, implication $(\gamma)\Rightarrow(\delta)$):
Let $R$ be a ring, $F$ be a free $R$-module and $K\subseteq F$ be an $R$-submodule. Assume that for every $x\in K$ there exists an $R$-module homomorphism $\upsilon:F\to K$ such that $\upsilon(x)=x$. Then for every finite sequence $x_1,\ldots,x_n$ of elements of $K$ there exists an $R$-module homomorphism $\upsilon:F\to K$ such that $\upsilon(x_i)=x_i$ for every $1\leq i\leq n$.
My try. I'm starting with the case $n=2$. By assumption there exists $\upsilon_1,\upsilon_2:F\to K$ such that $\upsilon_1(x_1)=x_1$ and $\upsilon_2(x_2)=x_2$. I'm failed to build $\upsilon$ as a linear combination of $\upsilon_1$ and $\upsilon_2$. Moreover, I cannot find a useful way to use assumption about $F$ to be a free $R$-module. Any hint will be appreciated.