Definition of total order and a claim Claim:  1) x $\in$ A will be  the smaller if to $a \in A$ there $xRa$
        2) y $\in$ A will be  the bigger if to $a \in A$ there $aRy$

the next question is about prove or give a claim against:

In total order, if term is the only minimal so he is the smaller.
  

my example was that if we will take $$R =\{ (x,y)\in \mathbb {Z}^2 | x\leq y \} $$
so we can see that with Hasse diagram its a infinite line. 
the claim againt were to add some $X$ that will be a minimal but is not the smaller, it can be $X$ that we added to $R$ let set $R \cup X_{0}$

Because you can prove and disprove this subject so I wanted to hear more arguments and see if this is true or not.
Thanks!
EDIT

 A: I think you'll be better understood if you say "the smallest" or "the minimum" instead of "the smaller." So, here's how I read your question:

Let $R$ be a total order on $A$. If $x$ is the only minimal element, that is, the only element for which there does not exist $y$ such that $rRx,$ then $x$ is the smallest element of $A$.

Your attempted counterexample does not work. For suppose we add some $X$ as a minimal element for the order $R$ on $\Bbb{Z}$. Then if $a\in\Bbb{Z}\cup\{X\}$ and $a\neq X$, $XRa,$ since $R$ is a total order and $aRX$ cannot hold. This same proof shows that any minimal element of for a total order is actually a least element, and so there can be only one minimal element.
This is an unexciting result, but things get more interesting in the case of partial orders: it's possible to construct partial orders with any number of minimal elements and with or without a least element. For example, we could take your order $R$ on $\Bbb{Z}$ and just add $X$ without relating it to any integers. Then $aRX$ doesn't hold for any $a\neq X,$ so $X$ is minimal, but there's still no element less than every other.
