On Tits Buildings of Type $A_n$

Let $P$ and $P^{\mathrm{\prime }}$ be projective spaces having the same dimension $n$ assumed to be finite. Let $\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{\mathrm{\prime }}$ be buildings of type $A_n$. We consider mappings of chambers of $\mathrm{\Delta }$ into chambers of ${\mathrm{\Delta }}^{\mathrm{\prime }}$ which send appartments to appartments. It is shown that shuch mappings are induced by strong embeddings of $P$ into $P^{\mathrm{\prime }}$ or into the dual space $P^{{}^{\mathrm{\prime }}\ast }$.