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 Th29:
  for GF being Field, V being VectSp of GF, a being Element of GF,
L being Linear_Combination of V st a <> 0.GF holds Carrier(a * L) = Carrier(L)
proof
  let GF be Field, V be VectSp of GF, a be Element of GF, L be
  Linear_Combination of V;
  set T = {u where u is Vector of V : (a * L).u <> 0.GF};
  set S = {v where v is Vector of V : L.v <> 0.GF};
  assume
A1: a <> 0.GF;
  T = S
  proof
    thus T c= S
    proof
      let x be object;
      assume x in T;
      then consider u being Vector of V such that
A2:   x = u and
A3:   (a * L).u <> 0.GF;
      (a * L).u = a * L.u by Def9;
      then L.u <> 0.GF by A3;
      hence thesis by A2;
    end;
    let x be object;
    assume x in S;
    then consider v being Vector of V such that
A4: x = v and
A5: L.v <> 0.GF;
    (a * L).v = a * L.v by Def9;
    then (a * L).v <> 0.GF by A1,A5,VECTSP_1:12;
    hence thesis by A4;
  end;
  hence thesis;
end;
