
theorem
  for L being non empty reflexive antisymmetric RelStr for a being
  Element of L holds sup {a} = a & inf {a} = a
proof
  let L be non empty reflexive antisymmetric RelStr;
  let 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 sup {a} = a by A1,Th30;
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,Th31;
end;
