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 Th28:
  Carrier(a * L) c= Carrier(L)
proof
  set T = {u : (a * L).u <> 0.GF};
  set S = {v : L.v <> 0.GF};
  T c= S
  proof
    let x be object;
    assume x in T;
    then consider u such that
A1: x = u and
A2: (a * L).u <> 0.GF;
    (a * L).u = a * L.u by Def9;
    then L.u <> 0.GF by A2;
    hence thesis by A1;
  end;
  hence thesis;
end;
