reserve p,q,r for FinSequence,
  x,y,y1,y2 for set,
  i,k for Element of NAT,
  GF for add-associative right_zeroed right_complementable Abelian associative
  well-unital distributive non empty doubleLoopStr,
  V for Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF,
  u,v,v1,v2,v3,w for Element of V,
  a,b for Element of GF,
  F,G ,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, GF;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem Th24:
  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 Def4;
  then
A1: Carrier(L1) \/ Carrier(L2) c= A by XBOOLE_1:8;
  Carrier(L1 + L2) c= Carrier(L1) \/ Carrier(L2) by Th23;
  hence Carrier(L1 + L2) c= A by A1;
end;
