Proposition 10.3. Assuming that:

$M$ is an $L$ -structure
$N^{*}$ an $L_{M}$ -structure such that $N^{*} ⊨ D (M)$
let $N$ be the $L$ -reduct of $N^{*}$ to $L$ (means throw away $L_{M} ∖ L$ sentences)
define $h : M \to N$ such that $h (a) = \underset{̲}{a} = a^{N}$ .

Then

h

is an

L

-embedding. Moreover, if

N^{*} ⊨ {T h}_{M} (M)

then

h

L