let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being Subspace of M holds
( ((0). M) /\ W = (0). M & W /\ ((0). M) = (0). M )
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W being Subspace of M holds
( ((0). M) /\ W = (0). M & W /\ ((0). M) = (0). M )
let W be Subspace of M; :: thesis: ( ((0). M) /\ W = (0). M & W /\ ((0). M) = (0). M )
0. M in W by VECTSP_4:17;
then 0. M in the carrier of W by STRUCT_0:def 5;
then {(0. M)} c= the carrier of W by ZFMISC_1:31;
then A1: {(0. M)} /\ the carrier of W = {(0. M)} by XBOOLE_1:28;
A2: the carrier of (((0). M) /\ W) = the carrier of ((0). M) /\ the carrier of W by Def2
.= {(0. M)} /\ the carrier of W by VECTSP_4:def 3 ;
hence ((0). M) /\ W = (0). M by A1, VECTSP_4:def 3; :: thesis: W /\ ((0). M) = (0). M
thus W /\ ((0). M) = (0). M by A2, A1, VECTSP_4:def 3; :: thesis: verum