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Let $E$ be a fixed real function $F$-space, i.e., $E$ is an order ideal in $L_0(S,\Sigma ,\mu )$ endowed with a monotone $F$-norm $\parallel \parallel$ under which $E$ is topologically complete. We prove that $E$ contains an isomorphic (topological) copy of $\omega$, the space of all sequences, if and only if $E$ contains a lattice-topological copy $W$ of $\omega$. If $E$ is additionally discrete, we obtain a much stronger result: $W$ can be a projection band; in particular, $E$ contains a complemented copy of $\omega$. This solves partially the open problem set recently by W....