reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem Th38:
  L1 is Linear_Combination of A & L2 is Linear_Combination of A
  implies L1 + L2 is Linear_Combination of A
proof
  assume L1 is Linear_Combination of A & L2 is Linear_Combination of A;
  then Carrier(L1) c= A & Carrier(L2) c= A by Def5;
  then
A1: Carrier(L1) \/ Carrier(L2) c= A by XBOOLE_1:8;
  Carrier(L1 + L2) c= Carrier(L1) \/ Carrier(L2) by Th37;
  hence Carrier(L1 + L2) c= A by A1;
end;
