reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve x,y for Point of X;
reserve A, B for Subset of X;
reserve P, Q for Subset of X;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace,
  Y0 for non empty SubSpace of X;

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;
