Throughout in the paper we assume that $M$ is an totally ordered structure.
{\bf Definition 1.} A partition $(A,B)$ of a model $M$ is said to be {\it cut} if $A < B$. Here, $A < B \iff \forall a \in A \; \forall b \in B \; (a < b).$ A cut is said to be {\it rational} if $A$ has maximal element or $B$ has minimal one, or one of them is empty. We say that cut is {\it quasirational} if $A$ and consequently $B$ are definable. A non-quasirational cut is said to be {\it irrational}. A model $M$ is said to be {\it Dedekind complete} if any its cut is rational. We say that model $M$ is {\it quasi Dedekind complete} if any its cut is quasirational. Let $M\prec N$ We say that the cut $(A,B)$ in $M$ is realized in $N$ if there exists $\alpha\in N\setminus M$ such that $A < \alpha < B.$\\ {\bf Definition 2.} [L. van den Dries, Pillay, Steinhorn] A totally ordered structure $M$ is said to be {\it o-minimal,} if every definable (with parameters) subset of $M$ is a finite union of points in $M$ and intervals $(a,b),\; a \in M \bigcup \{ -\infty \},\; b \in M \bigcup\{\infty\}$. \\ {\bf Definition 3.}[Marker-Steinhorn] Let $M \prec N \models T$, $T$ be an o-minimal theory. A model $M$ is to be said {\it Dedekind complete in $N$} if there is no non-rational cut in $M$ realized in $N$.\\ {\bf Definition 4.}[Dickmann, Marker-Macpherson-Steinhorn] A totally ordered structure $M$ is said to be {\it weakly o-minimal,} if every definable (with parameters) subset of $M$ is finite union of convex sets.
Theory $T$ is {\it weakly o-minimal} if every model of $T$ is weakly o-minimal.
{\bf Definition 5.} Let $M \prec N \models T$, $T$ be a weakly o-minimal (w.o.m.) theory. We say that $M$ is {\it quasi Dedekind complete in $N$} if there is no irrational cut in $M$ realized in $N$.\\ Lou van den Dries had studied definable types over real closed fields and proved the following result:\\ {\bf Theorem 1.} Every type over $(R,+,\cdot,0,1)$ is definable.\\ D.Marker and Ch.Steinhorn had strengthened this result expanding it for o-minimal theories:\\ {\bf Theorem 2.} Let $T$ be an o-minimal theory, $M \models T$. Then the following holds: \begin{enumerate} \item $M$ is Dedekind complete $\iff$ every $n$--type of $\bigcup_{n<\omega} S_n(M)$ is definable. \item Let $M \prec N$. Then $(\forall \bar \alpha \in N \setminus M)\; tp(\bar \alpha / M) $ is definable $\iff M $ is Dedekind complete in $N$. \end{enumerate} We prove the following theorem:\\ {\bf Theorem 3.} Let $M$ be a model of a w.o.m. theory $T$. Then $M$ is quasi Dedekind complete iff any $p(\bar v) \in \bigcup_{n<\omega} S_n(M)$ is definable.\\ And we also have the following:\\ {\bf Proposition 1.} There is a w.o.m. theory with two models $M$, $N$ such that $M\prec N$ and $\exists \bar\alpha\in N\setminus M$: $$tp(\bar\alpha/M) \mbox{ is non-definable } \mbox{ and } M \mbox{ is quasi Dedekind complete in }N.$$