of type $V_j$, then the residue $R(a)$ is $E_i$-connected for $i=1,\ldots n$.

Note that $\K'$ is closed under finitely generated substructures by the choice of language.

Let $A$ be a finite $L_n'$-graph which is $E_i$-connected for $i=1,\ldots n$. The following extensions are called elementary strong extensions of $A$:

1.  add a vertex of type $P$ or $E$ to $A$ which is connected to at most one vertex of $A$ and such that the extension is still $E_i$-connected for all $i=1,\ldots n$.

2.  If $(x,y,z)$ is a dense flag in $A$, add a vertex $y'$ of the same type as $z$ to $A$ such that $(x,y',z)$ is a flag.

3.  if $|A|\leq 1$ contains no line, add a vertex of appropriate type which is connected to the vertex of $A$ if $A\neq\emptyset$.

Again we write $A\leq B$ if $B$ arises from $A$ by finitely many elementary strong extensions.

We next show that $(\K_n',\leq)$ has the amalgamation property for $\leq$-extensions.

If $A$ contains a flag of type $(V_1,\ldots V_{n-1})$ and $A\leq B,C$ are in $\K_n'$, then $D:=B\otimes_AC\in\K_n'$ and $B,C\leq D$.

Clearly, $B,C\leq D$ and $D$ is $E_i$-connected for all $i=1,\ldots n$ since $A$ contains a flag. To see that $D\in\K$, note that 