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The article addresses the problem of finding a small unsatisfiable core of an unsatisfiable CNF formula. The proposed algorithm, CoreTrimmer, iterates over each internal node d in the resolution graph that `consumes' a large number of clauses M (i.e., a large number of original clauses are present in the unsat core with the sole purpose of proving d) and attempts to prove them without the M clauses. If this is possible, it transforms the resolution graph into a new graph that does not have the M clauses at its core. CoreTrimmer can be integrated into a fixpoint framework similarly to Malik and Zhang's fix-point algorithm run_till_ fix. We call this option trim_till_fix. Experimental evaluation on a large number of industrial CNF unsatisfiable formulas shows that trim_till_fix doubles, on average, the number of reduced clauses in comparison to run_till_fix. It is also better when used as a component in a bigger system that enforces short timeouts.