let V be RealLinearSpace; :: thesis: for W being Subspace of V holds (0). W = (0). V
let W be Subspace of V; :: thesis: (0). W = (0). V
( the carrier of ((0). W) = {(0. W)} & the carrier of ((0). V) = {(0. V)} ) by Def3;
then ( the carrier of ((0). W) = the carrier of ((0). V) & (0). W is Subspace of V ) by Def2, Th35;
hence (0). W = (0). V by Th38; :: thesis: verum