  In \cite{cartier2}, Pierre Cartier conjectured that for any non commutative
formal power series $\Phi$ on $X=\{x_0,x_1\}$ with coefficients in a
$\Q$-extension, $A$, subjected to some suitable conditions, there exists an
unique algebra homomorphism $\varphi$ from the $\Q$-algebra generated by the
convergent polyz\^etas to $A$ such that $\Phi$ is computed from $\Phi_{KZ}$
Drinfel'd associator by applying $\varphi$ to each coefficient. We prove
$\varphi$ exists and it is a free Lie exponential over $X$. Moreover, we give a
complete description of the kernel of polyz\^eta and draw some consequences
about a structure of the algebra of convergent polyz\^etas and about the
arithmetical nature of the Euler constant.
