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 W1, W2 being strict Subspace of M holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
let M be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W1, W2 being strict Subspace of M holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
let W1, W2 be strict Subspace of M; :: thesis: ( W1 + W2 = W2 iff W1 /\ W2 = W1 )
( ( W1 + W2 = W2 implies W1 is Subspace of W2 ) & ( W1 is Subspace of W2 implies W1 + W2 = W2 ) & ( W1 /\ W2 = W1 implies W1 is Subspace of W2 ) & ( W1 is Subspace of W2 implies W1 /\ W2 = W1 ) ) by Th12, Th21;
hence ( W1 + W2 = W2 iff W1 /\ W2 = W1 ) ; :: thesis: verum