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 Th30:
  S <> {} iff not S is Initial_Segm of S
proof
  thus S <> {} implies not S is Initial_Segm of S
  proof
    assume S <> {} & S is Initial_Segm of S;
    then consider a such that
A1: a in S & S = InitSegm(S,a) by Def11;
    a in LowerCone{a} by A1,XBOOLE_0:def 4;
    hence thesis by Th21;
  end;
  thus thesis by Def11;
end;
