 reserve x,y for object, X,Y,Z for set;
 reserve GF for commutative
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;
 reserve GF for commutative non degenerated almost_left_invertible
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;
reserve l0 for Linear_Combination of {}(the carrier of V);
reserve x for set,
  R for Ring,
  V for LeftMod of R,
  v,v1,v2 for Vector of V,
  A, B for Subset of V;

theorem 
  for R being domRing, V being LeftMod of R,
      L being Linear_Combination of V, a being Scalar of R holds
      a <> 0.R implies Carrier(a * L) = Carrier(L)
  proof
  let R be domRing, V be LeftMod of R, L be Linear_Combination of V, a be
  Scalar of R;
  set T = {u where u is Vector of V : (a * L).u <> 0.R};
  set S = {v where v is Vector of V : L.v <> 0.R};
  assume
A1: a <> 0.R;
  T = S
  proof
    thus T c= S
    proof
      let x be object;
      assume x in T;
      then consider u be Vector of V such that
A2:   x = u and
A3:   (a * L).u <> 0.R;
      (a * L).u = a * L.u by VECTSP_6:def 9;
      then L.u <> 0.R by A3;
      hence thesis by A2;
    end;
    let x be object;
    assume x in S;
    then consider v be Vector of V such that
A4: x = v and
A5: L.v <> 0.R;
    (a * L).v = a * L.v by VECTSP_6:def 9;
    then (a * L).v <> 0.R by A1,A5,VECTSP_2:def 1;
    hence thesis by A4;
  end;
  hence thesis;
end;
