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 W1, W2, W3 being Subspace of M st W1 is Subspace of W2 & W1 is Subspace of W3 holds
W1 is Subspace of W2 /\ W3
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 W1, W2, W3 being Subspace of M st W1 is Subspace of W2 & W1 is Subspace of W3 holds
W1 is Subspace of W2 /\ W3
let W1, W2, W3 be Subspace of M; :: thesis: ( W1 is Subspace of W2 & W1 is Subspace of W3 implies W1 is Subspace of W2 /\ W3 )
assume A1: ( W1 is Subspace of W2 & W1 is Subspace of W3 ) ; :: thesis: W1 is Subspace of W2 /\ W3

now :: thesis: for v being Element of M st v in W1 holds
v in W2 /\ W3

let v be Element of M; :: thesis: ( v in W1 implies v in W2 /\ W3 )
assume v in W1 ; :: thesis: v in W2 /\ W3
then ( v in W2 & v in W3 ) by A1, VECTSP_4:8;
hence v in W2 /\ W3 by Th3; :: thesis: verum

end;

hence W1 is Subspace of W2 /\ W3 by VECTSP_4:28; :: thesis: verum