let R be Ring; :: thesis: for V being RightMod of R holds {(0. V)} is linearly-closed
let V be RightMod of R; :: thesis: {(0. V)} is linearly-closed
thus for v, u being Vector of V st v in {(0. V)} & u in {(0. V)} holds
v + u in {(0. V)} :: according to RMOD_2:def 1 :: thesis: for a being Scalar of R
for v being Vector of V st v in {(0. V)} holds
v * a in {(0. V)}

proof

let v, u be Vector of V; :: thesis: ( v in {(0. V)} & u in {(0. V)} implies v + u in {(0. V)} )
assume ( v in {(0. V)} & u in {(0. V)} ) ; :: thesis: v + u in {(0. V)}
then ( v = 0. V & u = 0. V ) by TARSKI:def 1;
then v + u = 0. V by RLVECT_1:def 4;
hence v + u in {(0. V)} by TARSKI:def 1; :: thesis: verum

end;

let a be Scalar of R; :: thesis: for v being Vector of V st v in {(0. V)} holds
v * a in {(0. V)}
let v be Vector of V; :: thesis: ( v in {(0. V)} implies v * a in {(0. V)} )
assume A1: v in {(0. V)} ; :: thesis: v * a in {(0. V)}
then v = 0. V by TARSKI:def 1;
hence v * a in {(0. V)} by A1, VECTSP_2:32; :: thesis: verum