reserve A for RelStr;
reserve X for non empty set;
reserve PX,PY,PZ,Y,a,b,c,x,y for set;
reserve S1,S2 for Subset of Y;

theorem Th1:
  for a being set holds InclPoset {{a}} is non empty reflexive
  transitive antisymmetric with_superior with_comparable_down
proof
  let a be set;
  set A = {{a}};
  set R9 = RelIncl A;
  reconsider R = R9 as Relation of A;
  set L = RelStr(#A,R#);
A1: L is with_superior
  proof
    set max1 = {a};
    reconsider max9 = max1 as Element of L by TARSKI:def 1;
    take max9;
A2: for y be set st y in field R & y <> max9 holds [y,max9] in R
    proof
      let y be set such that
A3:   y in field R and
A4:   y <> max9;
      field R c= A \/ A by RELSET_1:8;
      hence thesis by A3,A4,TARSKI:def 1;
    end;
    [max9,max9] in R by WELLORD2:def 1;
    then max9 in field R by RELAT_1:15;
    hence thesis by A2,ORDERS_1:def 14;
  end;
A5: for x,y be Element of L holds (ex z be Element of L st z <= x & z <= y)
  implies x <= y or y <= x
  proof
    let x,y be Element of L;
    assume ex z be Element of L st z <= x & z <= y;
A6: y = {a} by TARSKI:def 1;
    x = {a} by TARSKI:def 1;
    then [x,y] in R by A6,WELLORD2:def 1;
    hence thesis by ORDERS_2:def 5;
  end;
  RelStr(#A,R#) = InclPoset A by YELLOW_1:def 1;
  hence thesis by A1,A5;
end;
