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Conceptual doubly lossy thin layers with high electric and magnetic conductances (HEMCs) are shown to exhibit perfect absorbing properties for a normally incident plane wave. In contrast to conventional absorbers and perfectly matched layers (PMLs), this absorber can be made infinitely thin in the limit of perfect electric and magnetic conductors. The dissipated power density within the HEMC is thus very high, though integrable. A straightforward derivation, backed by physical reasoning, serves to make this point.