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Resolution trees with lemmas (RTL) are a resolution-based propositional proof system that is related to the DPLL algorithm with clause learning. Its fragments $\ensuremath{\ensuremath{\mathrm{RTL}} ({k})}$ are related to clause learning algorithms where the width of learned clauses is bounded by k. For every k up to O(logn), an exponential separation between the proof systems $\ensuremath{\ensuremath{\mathrm{RTL}} ({k})}$ and $\ensuremath{\ensuremath{\mathrm{RTL}} ({k+1})}$ is shown.