reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;

theorem Th9:
  for A being non empty reflexive RelStr, a1,a2 being Element of A
  holds {a1,a2} is Chain of A iff a1 <= a2 or a2 <= a1
proof
  let A be non empty reflexive RelStr, a1,a2 be Element of A;
A1: a2 <= a2;
  thus {a1,a2} is Chain of A implies a1 <= a2 or a2 <= a1
  proof
    assume {a1,a2} is Chain of A;
    then
A2: the InternalRel of A is_strongly_connected_in {a1,a2} by Def7;
    a1 in {a1,a2} & a2 in {a1,a2} by TARSKI:def 2;
    then
    [a1,a2] in the InternalRel of A or [a2,a1] in the InternalRel of A by A2;
    hence thesis;
  end;
  assume
A3: a1 <= a2 or a2 <= a1;
A4: a1 <= a1;
  {a1,a2} is strongly_connected
  proof
    let x,y be object;
    assume
A5: x in {a1,a2} & y in {a1,a2};
    now
      per cases by A5,TARSKI:def 2;
      suppose
        x = a1 & y = a1;
        hence thesis by A4;
      end;
      suppose
        x = a1 & y = a2;
        hence thesis by A3;
      end;
      suppose
        x = a2 & y = a1;
        hence thesis by A3;
      end;
      suppose
        x = a2 & y = a2;
        hence thesis by A1;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
