
theorem Th42:
  for L being lower-bounded antisymmetric non empty RelStr holds
  ex_sup_of {},L & ex_inf_of the carrier of L, L
proof
  let L be lower-bounded antisymmetric non empty RelStr;
  consider a being Element of L such that
A1: a is_<=_than the carrier of L by Def4;
  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_sup_of {},L by Th15;
  for b being Element of L st the carrier of L is_>=_than b holds a >= b;
  hence thesis by A1,Th16;
end;
