reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, P being bounded Subset of X, Q being
  Subset of X st Q c= P holds Q is bounded
proof
  let X be LinearTopSpace, P be bounded Subset of X, Q being Subset of X such
  that
A1: Q c= P;
  let V be a_neighborhood of 0.X;
  consider s such that
A2: s > 0 and
A3: for t st t > s holds P c= t*V by Def12;
  take s;
  thus s > 0 by A2;
  let t;
  assume t > s;
  then P c= t*V by A3;
  hence thesis by A1;
end;
