In the paper it is shown that all properties of Robinson's finite forcing are naturally transmitted to a forcing whose set of conditions has as its elements finite sets of $\Sigma_n,\prod_n$ sentences of an extension of an original language (in wich the theory in question $T$ is defined) consistent with $T$. In fact we could speak of ``a translation for $n$'' of $fi$ the results of nite forcing (let us put $n=0$ and there they are). So, for a given $n$, the coresponding forcing property determines an $n$-companion operator (--)$f_n$ (definition 4.1) (the other such one is (--)$e_n$ (4.4)). Theory $Tf_n$ (associated with theory $T$) is $f_n$-complete (4.10) and the class of generic structures (we call them $T$--$n$ finitely generic) consists of exacdy those models $M$ of $Tj_n$ which ``$n$-complete'' $Tf_n$ (in other words from $M\prec{}_nN\,|=Tf_n$ follows $M\prec N$) (3.8) and, of course, if language $L$ is countable this class is axiomatizable if and only if $Tf_n$ is $n$-model complete (3.13). Furthermore, the $n$-model companion and $n$-model completion are defined in the only possible way. Certainly if $T$ has an $n$-companion operator (--)${}^*$ it is unique and $T^*=Th(E^n_T)=Tf_n$ (3.23, 3.24). The construction of $Tf_n$ by standard model's theoretical means is a known usage of Henrard's approximation chains [6]. Thus an application of such a forcing property (for $n>0$) could be of interest (if at all) in cases of theories which are not $\prod_2$ axiomatizable. Naturally on condition that we wish to have the given theory $T$ contained in the one associated with it and are ready to pay this by loosing the model completness of the class of finitely generic structures as well as some other ``nice'' properties of the (--)${}^f$ operator. In the text by $\Sigma_n$, $\prod_n$ sentences we mean sentences logically equivalent to those which are $\Sigma_n$, $\prod_n$ in the strong sense of definition (\S0). However this is of no importance i.e. there would not be any restriction if we had remained with the ``genuine'' ones. We used (without accepting them) the proofs of [1], [2], [5], [6] along with applying some well known model theoretical theorems as well as those of [3] which concern the general features of forcing properties (particularly 1.5); thus they are mostly either outliked or omitted. In \S3, after 3.7, in order to obtain other results we limited the fragment $K_A$ to the set of sentences of the language $K=L(C)$ and in \S4 all formulas and theories are defined in $L$.