let R be Ring; 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; 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; the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
let x be object ; TARSKI:def 3 ( 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))
; 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))
; verum