theorem Th31:
  L is Linear_Combination of A implies a * L is Linear_Combination of A
proof
  assume L is Linear_Combination of A;
  then
A1: Carrier(L) c= A by Def4;
  Carrier(a * L) c= Carrier(L) by Th28;
  then Carrier(a * L) c= A by A1;
  hence thesis by Def4;
end;
