I have read some article and the article says that by the following conditions, there exists a unique vector of $x=(x_{1},...,x_{n})$ that solves the system of $N$ equations without further explanation.
$f_{i}(x)=p_{i}$, $\forall$ $i \in \{1,2,...,n\}$, where $f_{i} : \mathbb R^{n} \to \mathbb R$ is $C^{\infty}$ function and $p_{i}$'s are positive constants($p_{i}>0$, $\forall i \in n$). In other words, this provides a system of $N$ equations in the $N$ unknown elements.
If
(1) $\lim_{x_{i}\to 0}f_{i}(x)=\infty>p_{i}$ $\forall$ $i \in \{1,2,...,n\}$,
(2) $\lim_{x_{i}\to \infty}f_{i}(x)=0<p_{i}$ $\forall$ $i \in \{1,2,...,n\}$ ,
(3) $\frac{df_{i}(x)}{dx_{i}}<0$, $\forall$ $i \in \{1,2,...,n\}$
(4) $\frac{df_{i}(x)}{dx_{j}}<0$, $\forall$ $i,j\;(i\neq j) \in \{1,2,...,n\}$.
(5) $\left|\frac{df_{i}(x)}{dx_{i}}\right|>\left|\frac{df_{i}(x)}{dx_{j}}\right|$, $\forall$ $i,j\;(i\neq j) \in \{1,2,...,n\}$.
Then is it obvious that there exists a unique vector of $x=(x_{1},...,x_{n})$ that solves the system of $N$ equations?
I think it is not obvious. I tried to prove by myself, but I do not know where to start the proof. If you can give any hints of this problem, I will appreciate it. Thank you in advance.