let V be Z_Module; :: thesis: for W1, W2 being Submodule of V holds the carrier of W1 c= the carrier of (W1 + W2)
let W1, W2 be Submodule of V; :: thesis: the carrier of W1 c= the carrier of (W1 + W2)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W1 or x in the carrier of (W1 + W2) )
set A = { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } ;
assume x in the carrier of W1 ; :: thesis: x in the carrier of (W1 + W2)
then reconsider v = x as Element of W1 ;
reconsider v = v as VECTOR of V by Th25;
A1: v = v + (0. V) by RLVECT_1:4;
( v in W1 & 0. V in W2 ) by Th33;
then x in { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } by A1;
hence x in the carrier of (W1 + W2) by Def14; :: thesis: verum