let V be Z_Module; :: thesis: for u, v being VECTOR of V
for W being Submodule of V st u in W & v in W holds
u + v in W
let u, v be VECTOR of V; :: thesis: for W being Submodule of V st u in W & v in W holds
u + v in W
let W be Submodule of V; :: thesis: ( u in W & v in W implies u + v in W )
reconsider VW = the carrier of W as Subset of V by Def9;
assume ( u in W & v in W ) ; :: thesis: u + v in W
then A1: ( u in the carrier of W & v in the carrier of W ) ;
VW is linearly-closed by Lm3;
then u + v in the carrier of W by A1;
hence u + v in W ; :: thesis: verum