theorem
  A c= B & B is linearly-independent implies A is linearly-independent
proof
  assume that
A1: A c= B and
A2: B is linearly-independent;
  let l be Linear_Combination of A;
  reconsider L = l as Linear_Combination of B by A1,VECTSP_6:4;
  assume Sum(l) = 0.V;
  then Carrier(L) = {} by A2;
  hence thesis;
end;
