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, W3 being Subspace of M
for W1 being strict Subspace of M st W1 is Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + 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 W2, W3 being Subspace of M
for W1 being strict Subspace of M st W1 is Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
let W2, W3 be Subspace of M; :: thesis: for W1 being strict Subspace of M st W1 is Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
let W1 be strict Subspace of M; :: thesis: ( W1 is Subspace of W3 implies W1 + (W2 /\ W3) = (W1 + W2) /\ W3 )
assume A1: W1 is Subspace of W3 ; :: thesis: W1 + (W2 /\ W3) = (W1 + W2) /\ W3
hence (W1 + W2) /\ W3 = (W1 /\ W3) + (W3 /\ W2) by Lm13, VECTSP_4:29
.= W1 + (W2 /\ W3) by A1, Th16 ;
:: thesis: verum