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 W2, W1, W3 being Subspace of M holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let M be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W2, W1, W3 being Subspace of M holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let W2, W1, W3 be Subspace of M; :: thesis: the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of (W2 + (W1 /\ W3)) or x in the carrier of ((W1 + W2) /\ (W2 + W3)) )
assume x in the carrier of (W2 + (W1 /\ W3)) ; :: thesis: x in the carrier of ((W1 + W2) /\ (W2 + W3))
then x in { (u + v) where u, v is Element of M : ( u in W2 & v in W1 /\ W3 ) } by Def1;
then consider u, v being Element of M such that
A1: x = u + v and
A2: u in W2 and
A3: v in W1 /\ W3 ;
( v in W1 & v in W3 & x = v + u ) by A1, A3, Th7;
then ( x in { (v1 + v2) where v1, v2 is Element of M : ( v1 in W1 & v2 in W2 ) } & x in { (u1 + u2) where u1, u2 is Element of M : ( u1 in W2 & u2 in W3 ) } ) by A2;
then ( x in the carrier of (W1 + W2) & x in the carrier of (W2 + W3) ) by Def1;
then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by XBOOLE_0:def 4;
hence x in the carrier of ((W1 + W2) /\ (W2 + W3)) by Def2; :: thesis: verum