The wrench-closure workspace (WCW) of six-degree-of-freedom (DOF) parallel cable-driven mechanisms is defined as the set of poses of the moving platform of the mechanism for which any external wrench can be balanced by tension forces in the cables. This workspace is fundamental in order to analyze and design parallel cable-driven mechanisms. This paper deals with the class of six-DOF mechanisms driven by seven cables. Two theorems, which provide efficient means to test whether a given pose of the moving platform belongs to the WCW, are proposed. One of these two theorems reveals the nature of the boundary of the constant-orientation cross sections of the WCW. Moreover, some of the possible applications of these theorems are discussed and illustrated.