let S be Subset of V; :: thesis: ( S = {(0. V)} implies S is linearly-closed )
assume A1: S = {(0. V)} ; :: thesis: S is linearly-closed
thus for v, u being VECTOR of V st v in S & u in S holds
v + u in S :: according to ZMODUL01:def 8 :: thesis: for a being Integer
for v being VECTOR of V st v in S holds
a * v in S

proof

let v, u be VECTOR of V; :: thesis: ( v in S & u in S implies v + u in S )
assume ( v in S & u in S ) ; :: thesis: v + u in S
then ( v = 0. V & u = 0. V ) by A1, TARSKI:def 1;
then v + u = 0. V by RLVECT_1:4;
hence v + u in S by A1, TARSKI:def 1; :: thesis: verum

end;

let a be Integer; :: thesis: for v being VECTOR of V st v in S holds
a * v in S
let v be VECTOR of V; :: thesis: ( v in S implies a * v in S )
assume A2: v in S ; :: thesis: a * v in S
then v = 0. V by A1, TARSKI:def 1;
hence a * v in S by A2, Th1; :: thesis: verum