let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for u, v being VECTOR of V st u in W & v in W holds
u + v in W
let W be Subspace of V; :: thesis: for u, v being VECTOR of V st u in W & v in W holds
u + v in W
reconsider VW = the carrier of W as Subset of V by Def1;
let u, v be VECTOR of V; :: thesis: ( u in W & v in W implies u + v in W )
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 Lm1;
then u + v in the carrier of W by A1, RLSUB_1:def 1;
hence u + v in W ; :: thesis: verum