
theorem Th38:
  for L being non empty reflexive antisymmetric RelStr for a being
  Element of L holds ex_sup_of {a},L & ex_inf_of {a},L
proof
  let L be non empty reflexive antisymmetric RelStr, a be Element of L;
A1: for b being Element of L st b is_>=_than {a} holds b >= a by Th7;
A2: a <= a;
  then a is_>=_than {a} by Th7;
  hence ex_sup_of {a},L by A1,Th15;
A3: for b being Element of L st b is_<=_than {a} holds b <= a by Th7;
  a is_<=_than {a} by A2,Th7;
  hence thesis by A3,Th16;
end;
