let R be Ring; :: thesis: for V being LeftMod of R
for W1, W2, W3 being Submodule of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let V be LeftMod of R; :: thesis: for W1, W2, W3 being Submodule of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
let W1, W2, W3 be Submodule of V; :: 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 Vector of V : ( u in W2 & v in W1 /\ W3 ) } by VECTSP_5:def 1;
then consider v, u being Vector of V such that
A1: ( x = u + v & u in W2 ) and
A2: v in W1 /\ W3 ;
v in W3 by A2, Th94;
then x in { (u1 + u2) where u2, u1 is Vector of V : ( u1 in W2 & u2 in W3 ) } by A1;
then A3: x in the carrier of (W2 + W3) by VECTSP_5:def 1;
v in W1 by A2, Th94;
then x in { (v1 + v2) where v2, v1 is Vector of V : ( v1 in W1 & v2 in W2 ) } by A1;
then x in the carrier of (W1 + W2) by VECTSP_5:def 1;
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 VECTSP_5:def 2; :: thesis: verum