let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for M being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of 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 VectSp-like Abelian add-associative right_zeroed VectSpStr of 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 )
A1: 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 ;
0. M in W by VECTSP_4:25;
then 0. M in the carrier of W by STRUCT_0:def 5;
then {(0. M)} c= the carrier of W by ZFMISC_1:37;
then A2: ( {(0. M)} /\ the carrier of W = {(0. M)} & the carrier of ((0). M) = {(0. M)} ) by VECTSP_4:def 3, XBOOLE_1:28;
hence ((0). M) /\ W = (0). M by A1, VECTSP_4:37; :: thesis: W /\ ((0). M) = (0). M
thus W /\ ((0). M) = (0). M by A1, A2, VECTSP_4:37; :: thesis: verum