reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem Th42:
  Carrier(L * a) c= Carrier(L)
proof
  set T = {u : (L * a).u <> 0.R};
  set S = {v : L.v <> 0.R};
  T c= S
  proof
    let x be object;
    assume x in T;
    then consider u such that
A1: x = u and
A2: (L * a).u <> 0.R;
    (L * a).u = L.u * a by Def10;
    then L.u <> 0.R by A2;
    hence thesis by A1;
  end;
  hence thesis;
end;
