theorem
  for X being non empty real-membered set, Y being real-membered set st
  X c= Y & Y is bounded_below holds lower_bound Y <= lower_bound X
proof
  let X be non empty real-membered set, Y be real-membered set;
  assume X c= Y & Y is bounded_below;
  then t in X implies t >= lower_bound Y by Def2;
  hence thesis by Th43;
end;
