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 Th29:
  for I being Initial_Segm of S holds I c= S
proof
  let I be Initial_Segm of S;
  per cases;
  suppose
    S = {};
    hence thesis by Def11;
  end;
  suppose
    S <> {};
    then ex a st a in S & I = InitSegm(S,a) by Def11;
    hence thesis by XBOOLE_1:17;
  end;
end;
