  Given a subgroup $\Gamma$ of rational points on an elliptic curve $E$ defined
over ${\mathbf Q}$ of rank $r \ge 1$ and any sufficiently large $x \ge 2$,
assuming that the rank of $\Gamma$ is less than $r$, we give upper and lower
bounds on the canonical height of a rational point $Q$ which is not in the
group $\Gamma$ but belongs to the reduction of $\Gamma$ modulo every prime $p
\le x$ of good reduction for $E$.
