let V be RealLinearSpace; :: thesis: for W1, W2 being strict Subspace of V holds

( W1 + W2 = W2 iff W1 /\ W2 = W1 )

let W1, W2 be strict Subspace of V; :: thesis: ( W1 + W2 = W2 iff W1 /\ W2 = W1 )

( W1 + W2 = W2 iff W1 is Subspace of W2 ) by Th8;

hence ( W1 + W2 = W2 iff W1 /\ W2 = W1 ) by Th17; :: thesis: verum

( W1 + W2 = W2 iff W1 /\ W2 = W1 )

let W1, W2 be strict Subspace of V; :: thesis: ( W1 + W2 = W2 iff W1 /\ W2 = W1 )

( W1 + W2 = W2 iff W1 is Subspace of W2 ) by Th8;

hence ( W1 + W2 = W2 iff W1 /\ W2 = W1 ) by Th17; :: thesis: verum