$\G$, then $\G\v_{AF2} t:\q xA$.

-   \(5) If $\G\v_{AF2} t:\q xA$, then, for every term $u$, $\G\v_{AF2} t:A[u/x]$.

-   \(6) If $\G\v_{AF2} t:A$, and $X$ is not free in $\G$, then $\G\v_{AF2} t:\q XA$.

-   \(7) If $\G\v_{AF2} t:\q XA$, then, for every formulas $G$, $\G\v_{AF2} t:A[G/X]$.

-   \(8) If $\G\v_{AF2} t:A[u/x]$, and $u \approx v$, then $\G\v_{AF2}
    t:A[v/x]$.

This typed $\l$-calculus system is called $AF2$ (for Arithmétique Fonctionnelle du second ordre).

(see \[2\]) The $AF2$ type system has the following properties :\
1) Type is preserved during reduction.\
2) Typable $\l$-terms are strongly normalizable.

We present now a syntaxical property of system $AF2$ that we will use afterwards.

(see \[8\]) If in the typing we go from $\G\v_{AF2} t:A$ to $\G\v_{AF2} t:B$, then we may assume that we begin by the $\q$-elimination rules, then by the equationnal rule, and finally by the $\q$-introduction rules.

-   We define on the set of types the two binary relations $\lhd$ and $\approx$ as the least reflexive and transitive binary relations such that :

    -   - $\q xA \lhd A[u/x]$, if $u$ is a term of language ;

    -   - $\q XA \lhd A[F/X]$, if $F$ is a formula of language ;

    -   