The theorem of Hanna Neumann ([{\bf 4}]) states that all associative operations $w(x,y)$ in the case of a free $G$ are of one of the following forms: $$ a, x, y, xay, yax, $$ where a is an arbitrary element of $G$. In the first part of this article we generalize this result. Theorem 1 shows that operations of the forms listed above are the only possible (except trivial cases) when we require $w(x,y)$ to satisfy not the associativity law, but any consequence of it (any weakened associativity law). In the second part of the article we determine all associative operations $w(x,y)$ in the case of $G$ free nilpotent of class two.