To facilitate the lifting of slab-like pieces (24) of material separated from a block (22) of material by substantially horizontal cuts (23), with the aid of a movable suction device (27) acting upon the upper surface of an upper slab-like piece of material, a pressurized gaseous medium is caused to act in the region of the cut (23) between the upper piece (24) of material and the underlying slab-like piece of material or residual part (22') of the block of material, either prior to and/or while lifting said upper piece. The pressurized medium is supplied into at least one cavity which has an open side and is formed in a movable cover means (54) arranged along a substantially vertical side of the block (22) subsequent to moving the cover means (54) to a position in which it is sealingly connected to and each open cavity side thereof is covered by a portion of said block side, which portion extends along and reaches above and below said region of the cut (23).