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;
