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 being Subspace of M
for W2 being strict Subspace of M holds
( W1 is Subspace of W2 iff W1 + W2 = 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 W1 being Subspace of M
for W2 being strict Subspace of M holds
( W1 is Subspace of W2 iff W1 + W2 = W2 )
let W1 be Subspace of M; :: thesis: for W2 being strict Subspace of M holds
( W1 is Subspace of W2 iff W1 + W2 = W2 )
let W2 be strict Subspace of M; :: thesis: ( W1 is Subspace of W2 iff W1 + W2 = W2 )
thus ( W1 is Subspace of W2 implies W1 + W2 = W2 ) :: thesis: ( W1 + W2 = W2 implies W1 is Subspace of W2 )

proof

assume W1 is Subspace of W2 ; :: thesis: W1 + W2 = W2
then the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
hence W1 + W2 = W2 by Lm3; :: thesis: verum

end;

thus ( W1 + W2 = W2 implies W1 is Subspace of W2 ) by Th7; :: thesis: verum