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 holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + 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 holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let W1, W2, W3 be Subspace of M; :: thesis: the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) or x in the carrier of (W2 /\ (W1 + W3)) )
assume x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) ; :: thesis: x in the carrier of (W2 /\ (W1 + W3))
then x in { (u + v) where v, u is Element of M : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by Def1;
then consider v, u being Element of M such that
A1: x = u + v and
A2: ( u in W1 /\ W2 & v in W2 /\ W3 ) ;
( u in W2 & v in W2 ) by A2, Th3;
then A3: x in W2 by A1, VECTSP_4:20;
( u in W1 & v in W3 ) by A2, Th3;
then x in W1 + W3 by A1, Th1;
then x in W2 /\ (W1 + W3) by A3, Th3;
hence x in the carrier of (W2 /\ (W1 + W3)) by STRUCT_0:def 5; :: thesis: verum