Proof
(related to Lemma: Gaussian Lemma (Number Theory))
 By hypothesis, $p > 2$ is an odd prime number and $n$ is an integer not divisible by $p.$
 First of all, we observe that according to creation of complete residue systems from others the congruence classes $ n,2n,3n,\dots ,{\frac {p1}{2}}n\mod p$ are all distinct and $ > 0$ as well as $ < p.$
 Therefore, indeed, there are $\frac {p1}2$ of them.
 Let $m\ge 0$ be the number of residues $ > \frac p2.$
 If $m > 0,$ let $b_1,\ldots,b_m$ denote the residues $ > \frac p2.$
 Correspondingly, by setting $l:=\frac {p1}2m,$ which is the number of residues $ < \frac p2,$ we denote them by $a_1,\ldots,a_l.$
 Multiplying all $\frac{p1}2$ residues with each other, we get the congruence^{1} $$\prod_{s=1}^l a_s\prod_{t=1}^m b_t\equiv \prod_{h=1}^{\frac{p1}2}hn\equiv \left(\frac{p1}2\right)!\;\;n^{\frac{p1}2}\mod p.$$
 We now observe that the numbers $pb_t$ lie all between $1$ and $\frac {p1}2$ and all these differences are distinct, since all $b_t$ are distinct.
 Moreover, each $a_s$ is different from each $pb_t.$ We prove this result by contradiction:
 Assume $a_s=pb_t$ for some indices $s,t.$
 Then there would be integers $x,y$ with $1\le x,y\le\frac{p1}2$ and $(xn)(p)\equiv (pyn)(p).$
 From this, it would follow $(xn)(p)\equiv (yn)(p),$ or $x(p)\equiv y(p),$ by congruence modulo a divisor.
 But $p\mid (x+y)$ contradicts $0 < x+y < p.$
 It follows that $a_s$ and $pb_t$ are disjoint for all indices $s,t$. In other words, they form together the numbers $1,\ldots,\frac{p1}2$ (possibly reordered).
 Therefore, we have
$$\left(\frac{p1}2\right)!\equiv \prod_{s=1}^l a_s\prod_{t=1}^m (pb_t)\equiv (1)^m\prod_{s=1}^l a_s\prod_{t=1}^m b_t\equiv (1)^m\left(\frac{p1}2\right)!\;\;n^{\frac{p1}2}\mod p.$$
 By congruence modulo a divisor, we get finally the congruence $$1\equiv (1)^m\cdot n^{\frac{p1}2}\mod p.$$
 The Euler's criterion for quadratic residues gives us $\left(\frac np\right)\equiv(1)^m\mod p.$
 Therefore, $\left(\frac np\right)=(1)^m,$ for the Legendre symbol $\left(\frac np\right),$ as required.
∎
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References
Bibliography
 Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
Footnotes