Theorem 2

Suppose that in the ring $K$ the relation $n.1 = 0$ implies $n = 0$ for every integer $n$. Let $r$ be a mapping of $X$ into $\hat{A}(X)$ such that $\omega(r(x)) \geq 2$ for $x \in X$ and let $g$ be the homomorphism of $F(X)$ into the Magnus group $\Gamma(X)$ such that $g(x) = 1 + x + r(x)$ for $x \in X$. For all $n \geq 1$, $C^n F(X)$ is the inverse image under $g$ of the subgroup $1 + \hat{A}_n(X)$ of $\Gamma(X)$.