Corollary 1

There exists on the algebra $A(X)$ a unique coproduct making $A(X)$ into a bigebra such that the elements of X are primitive. Further, $\beta$ is an isomorphism of the bigebra $U(L(X))$ onto $A(X)$ with this bigebra structure.

This follows from assertion (b) of the theorem and the fact that X generates the unital algebra $A(X)$.

Henceforth $A(X)$ is given this bigebra structure and $L(X)$ is identified with its image under $\alpha$, that is with the Lie subalgebra of $A(X)$ generated by X.