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 strict Subspace of M holds
( ((Omega). M) /\ W = W & W /\ ((Omega). M) = W )
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 strict Subspace of M holds
( ((Omega). M) /\ W = W & W /\ ((Omega). M) = W )
let W be strict Subspace of M; :: thesis: ( ((Omega). M) /\ W = W & W /\ ((Omega). M) = W )
A1: ( the carrier of (((Omega). M) /\ W) = the carrier of ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) /\ the carrier of W & the carrier of W c= the carrier of M ) by Def2, VECTSP_4:def 2;
hence ((Omega). M) /\ W = W by VECTSP_4:29, XBOOLE_1:28; :: thesis: W /\ ((Omega). M) = W
thus W /\ ((Omega). M) = W by A1, VECTSP_4:29, XBOOLE_1:28; :: thesis: verum