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 W2 being Subspace of M holds
( ( for W1 being strict Subspace of M st W1 is Subspace of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Subspace of M st W1 /\ W2 = W1 holds
W1 is Subspace of W2 ) )
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 W2 being Subspace of M holds
( ( for W1 being strict Subspace of M st W1 is Subspace of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Subspace of M st W1 /\ W2 = W1 holds
W1 is Subspace of W2 ) )
let W2 be Subspace of M; :: thesis: ( ( for W1 being strict Subspace of M st W1 is Subspace of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Subspace of M st W1 /\ W2 = W1 holds
W1 is Subspace of W2 ) )
thus for W1 being strict Subspace of M st W1 is Subspace of W2 holds
W1 /\ W2 = W1 :: thesis: for W1 being Subspace of M st W1 /\ W2 = W1 holds
W1 is Subspace of W2

proof

let W1 be strict Subspace of M; :: thesis: ( W1 is Subspace of W2 implies W1 /\ W2 = W1 )
assume W1 is Subspace of W2 ; :: thesis: W1 /\ W2 = W1
then A1: the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def2;
hence W1 /\ W2 = W1 by A1, VECTSP_4:29, XBOOLE_1:28; :: thesis: verum

end;

thus for W1 being Subspace of M st W1 /\ W2 = W1 holds
W1 is Subspace of W2 by Th15; :: thesis: verum