reserve  X for non empty set,
  R for Relation of X;

theorem
  for L being non empty reflexive antisymmetric RelStr for x,y being
  Element of L holds x <= y implies sup {x,y} = y & inf {x,y} = x
proof
  let R be non empty reflexive antisymmetric RelStr;
  let a,b be Element of R;
A1: for x being Element of R st x is_>=_than {a,b} holds b <= x
  proof
    let a0 be Element of R;
A2: b in {a,b} by TARSKI:def 2;
    assume a0 is_>=_than {a,b};
    hence thesis by A2,LATTICE3:def 9;
  end;
A3: for x being Element of R st x is_<=_than {a,b} holds a >= x
  proof
    let a0 be Element of R;
A4: a in {a,b} by TARSKI:def 2;
    assume a0 is_<=_than {a,b};
    hence thesis by A4,LATTICE3:def 8;
  end;
  assume
A5: a <= b;
  for x being Element of {a,b} holds x >= a
  proof
    let a0 be Element of {a,b};
    a <= a0 or a <= a0 by A5,TARSKI:def 2;
    hence thesis;
  end;
  then for x being Element of R st x in {a,b} holds x >= a;
  then
A6: a is_<=_than {a,b} by LATTICE3:def 8;
  for x being Element of R st x in {a,b} holds x <= b by A5,TARSKI:def 2;
  then b is_>=_than {a,b} by LATTICE3:def 9;
  hence thesis by A6,A1,A3,YELLOW_0:30,31;
end;
