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 (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (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 W1, W2, W3 being Subspace of M holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let W1, W2, W3 be Subspace of M; :: thesis: the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let x be object ; :: 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 v, u is Element of M : ( u in W2 & v in W1 /\ W3 ) } by Def1;
then consider v, u being Element of M such that
A1: ( x = u + v & u in W2 ) and
A2: v in W1 /\ W3 ;
v in W3 by A2, Th3;
then x in { (u1 + u2) where u2, u1 is Element of M : ( u1 in W2 & u2 in W3 ) } by A1;
then A3: x in the carrier of (W2 + W3) by Def1;
v in W1 by A2, Th3;
then x in { (v1 + v2) where v2, v1 is Element of M : ( v1 in W1 & v2 in W2 ) } by A1;
then x in the carrier of (W1 + W2) by Def1;
then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by A3, XBOOLE_0:def 4;
hence x in the carrier of ((W1 + W2) /\ (W2 + W3)) by Def2; :: thesis: verum