let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of W1 c= the carrier of (W1 + W2)
let V be RightMod of R; :: 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 set ; :: 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 v, u 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 RMOD_2:18;
( v in W1 & 0. V in W2 & v = v + (0. V) ) by RLVECT_1:def 7, RMOD_2:25, STRUCT_0:def 5;
then x in { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) } ;
hence x in the carrier of (W1 + W2) by Def1; :: thesis: verum