Corollary 3

Let $f:G\to H$ be a homomorphism of groups with operators, $N$ its kernel, $X$ a subset of $G$ such that $f(X)$ generates $H$ and $Y$ a subset of $N$ which generates $N$. Then $X\cup Y$ generates $N$.

Let $G'$ be the stable subgroup of $G$ generated by $X\cup Y$. As $Y\subset G'$, $N\subset G'$.

As $f(X)\subset f(G')$, $f(G')=H$, whence $G'=f^{-1}(H)=G$.