let R be Ring; :: thesis: for V being RightMod 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 RightMod 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 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 Vector of V : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by Def1;
then consider u, v being Vector of V 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, RMOD_2:28;
then x in W2 /\ (W1 + W3) by Th7;
hence x in the carrier of (W2 /\ (W1 + W3)) by STRUCT_0:def 5; :: thesis: verum