  Szpilrajn's Lemma entails that each partial order extends to a linear order.
Dushnik and Miller use Szpilrajn's Lemma to show that each partial order has a
relizer. Since then, many authors utilize Szpilrajn's Theorem and the
Well-ordering principle to prove more general existence type theorems on
extending binary relations. Nevertheless, we are often interested not only in
the existence of extensions of a binary relation $R$ satisfying certain axioms
of orderability, but in something more: (A) The conditions of the sets of
alternatives and the properties which $R$ satisfies to be inherited when one
passes to any member of a subfamily of the family of extensions of $R$ and: (B)
The size of a family of ordering extensions of $R$, whose intersection is $R$,
to be the smallest one. The key to addressing these kinds of problems is the
szpilrajn inherited method. In this paper, we define the notion of
$\Lambda(m)$-consistency, where $m$ can reach the first infinite ordinal
$\omega$, and we give two general inherited type theorems on extending binary
relations, a Szpilrajn type and a Dushnik-Miller type theorem, which generalize
all the well known existence and inherited type extension theorems in the
literature. \keywords{Consistent binary relations, Extension theorems,
Intersection of binary relations.
