Theorem: Let y be a quadratic residue in Z_N* where N=pq. There are exactly four integers x_1, x_2, x_3 and x_4 where 0 < x_1 < x_2 < N/2 < x_3 < x_4 < N such that y = x_i^2 mod N for i =1,2,3,4.

The above theorem simply say that the exactly two of the four roots must be greater than N/2.

Most paper will state that this result is well known, without provide any detail proof. How can we proof 0 < x_1 < x_2 < N/2 < x_3 < x_4 < N ?

Theorem: Let $y$ be a quadratic residue in $\mathbb{Z}_N$* where $N=pq$. 

There are exactly four integers $x_1, x_2, x_3, x_4$ where $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$ such that $y = x_i^2 \pmod{N}$ for $i=1,2,3,4$.

The above theorem simply states that exactly two of the four roots must be greater than $\frac{N}{2}$.

Most papers will say that this result is well known, without providing any detailed proof. How can we prove that $0 < x_1 < x_2 < \frac{N}{2} < x_3 < x_4 < N$?