let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
let W1, W2 be Submodule of V; :: thesis: the carrier of ((W1 /\ W2) + W2) = the carrier of W2
thus the carrier of ((W1 /\ W2) + W2) c= the carrier of W2 :: according to XBOOLE_0:def 10 :: thesis: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 /\ W2) + W2) or x in the carrier of W2 )
assume x in the carrier of ((W1 /\ W2) + W2) ; :: thesis: x in the carrier of W2
then x in { (u + v) where u, v is Vector of V : ( u in W1 /\ W2 & v in W2 ) } by Def1;
then consider u, v being Vector of V such that
A1: x = u + v and
A2: u in W1 /\ W2 and
A3: v in W2 ;
u in W2 by A2, Th7;
then u + v in W2 by A3, RMOD_2:28;
hence x in the carrier of W2 by A1, STRUCT_0:def 5; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W2 or x in the carrier of ((W1 /\ W2) + W2) )
assume A4: x in the carrier of W2 ; :: thesis: x in the carrier of ((W1 /\ W2) + W2)
the carrier of W2 c= the carrier of (W2 + (W1 /\ W2)) by Lm2;
then the carrier of W2 c= the carrier of ((W1 /\ W2) + W2) by Lm1;
hence x in the carrier of ((W1 /\ W2) + W2) by A4; :: thesis: verum