reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem Th44:
  for X,Y being ext-real-membered set st X c= Y & Y is
  bounded_below holds X is bounded_below
proof
  let X,Y be ext-real-membered set;
  assume
A1: X c= Y;
  given r being Real such that
A2: r is LowerBound of Y;
  take r;
  thus thesis by A1,A2,Th5;
end;
