TITLE: Proving that $R_1 = R_2$ (linearly ordered sets)
QUESTION [0 upvotes]: Let $R_1$ and $R_2$ be linearly ordered sets in set $X$.
Prove that if $R_1R_2$ is linearly ordered set, then $R_1 = R_2$
I understand the defnitions of $R_1R_2$ and totally ordered sets, but when I start to prove, it seems that I still lack some knowledge. Definitions don't seem to be enough.

REPLY [0 votes]: $xR_{1}R_{2}z \iff  xR_{2}y$  and $yR_{1}z$ for some $y$. 
Assume that $xR_{2}z$. It is to be shown that $xR_{1}z$ and since $R_{1}$ is linear we have $xR_{1}z$ or $zR_{1}x$. In the first case we are ready. 
If $zR_{1}x$ then we combine this with $zR_{2}z$ and find $zR_{1}R_{2}x$.Next to that we have $xR_{2}z$ and $zR_{1}z$ leading to $xR_{1}R_{2}z$. Then $x=z$ and consequently $xR_{1}z$.
Proved is now that $R_{2}\subset R_{1}$ and likewise it can be shown that $R_{1}\subset R_{2}$.
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