theorem Th45:
  L is Linear_Combination of A implies L * a is Linear_Combination of A
proof
  assume L is Linear_Combination of A;
  then
A1: Carrier(L) c= A by Def5;
  Carrier(L * a) c= Carrier(L) by Th42;
  then Carrier(L * a) c= A by A1;
  hence thesis by Def5;
end;
