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 Th23:
  Carrier(L1 + L2) c= Carrier(L1) \/ Carrier(L2)
proof
  let x be object;
  assume x in Carrier(L1 + L2);
  then consider u such that
A1: x = u and
A2: (L1 + L2).u <> 0.GF;
  (L1 + L2).u = L1.u + L2.u by Th22;
  then L1.u <> 0.GF or L2.u <> 0.GF by A2,RLVECT_1:4;
  then x in {v1 : L1.v1 <> 0.GF} or x in {v2 : L2.v2 <> 0.GF} by A1;
  hence thesis by XBOOLE_0:def 3;
end;
