let R be Ring; :: thesis: for V being LeftMod of R
for W1, W2, W3 being Submodule of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let V be LeftMod of R; :: thesis: for W1, W2, W3 being Submodule of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let W1, W2, W3 be Submodule of V; :: 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 Vector of V : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by VECTSP_5:def 1;
then consider v, u being Vector of V such that
A1: x = u + v and
A2: ( u in W1 /\ W2 & v in W2 /\ W3 ) ;
( u in W2 & v in W2 ) by A2, Th94;
then A3: x in W2 by A1, Th36;
( u in W1 & v in W3 ) by A2, Th94;
then x in W1 + W3 by A1, Th92;
then x in W2 /\ (W1 + W3) by A3, Th94;
hence x in the carrier of (W2 /\ (W1 + W3)) ; :: thesis: verum