theorem Th44:
  L is Linear_Combination of A implies a * L is Linear_Combination of A
proof
  assume
A1: L is Linear_Combination of A;
  now
    per cases;
    suppose
      a = 0;
      then a * L = ZeroLC(V) by Th43;
      hence thesis by Th22;
    end;
    suppose
      a <> 0;
      then Carrier(a * L) = Carrier(L) by Th42;
      hence thesis by A1,Def6;
    end;
  end;
  hence thesis;
end;
