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, 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 VectSp-like Abelian add-associative right_zeroed VectSpStr of 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 set ; :: 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 u, v is Element of M : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by Def1;
then consider u, v being Element of M such that
A1: x = u + v and
A2: ( u in W1 /\ W2 & v in W2 /\ W3 ) ;
( u in W1 & u in W2 & v in W2 & v in W3 ) by A2, Th7;
then ( x in W1 + W3 & x in W2 ) by A1, Th5, VECTSP_4:28;
then x in W2 /\ (W1 + W3) by Th7;
hence x in the carrier of (W2 /\ (W1 + W3)) by STRUCT_0:def 5; :: thesis: verum