theorem
  for Y0 being SubSpace of Y, A being non empty Subset of Y holds A is
  Subset of Y0 implies Sspace(A) is SubSpace of Y0
proof
  let Y0 be SubSpace of Y, A be non empty Subset of Y;
  assume A is Subset of Y0;
  then A c= the carrier of Y0;
  then the carrier of Sspace(A) c= the carrier of Y0 by Lm3;
  hence thesis by Lm2;
end;
