let V be ComplexLinearSpace; :: 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 CLVECT_1:def 7 :: thesis: for z being Complex
for v being VECTOR of V st v in {(0. V)} holds
z * v 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_1:4;
hence v + u in {(0. V)} by TARSKI:def 1; :: thesis: verum

end;

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