
theorem Th43:
  for L being upper-bounded antisymmetric non empty RelStr holds
  ex_inf_of {},L & ex_sup_of the carrier of L, L
proof
  let L be upper-bounded antisymmetric non empty RelStr;
  consider a being Element of L such that
A1: a is_>=_than the carrier of L by Def5;
  now
    take a;
    thus a is_<=_than {};
    thus for b being Element of L st b is_<=_than {} holds a >= b by A1;
  end;
  hence ex_inf_of {},L by Th16;
  for b being Element of L st the carrier of L is_<=_than b holds a <= b;
  hence thesis by A1,Th15;
end;
