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 Th31:
  S <> {} & S is Initial_Segm of T implies not T is Initial_Segm of S
proof
  assume that
A1: S <> {} and
A2: S is Initial_Segm of T and
A3: T is Initial_Segm of S;
  now
    per cases;
    suppose
      S = {};
      hence thesis by A1;
    end;
    suppose
      T = {};
      hence thesis by A1,A2,Def11;
    end;
    suppose
A4:   S <> {} & T <> {};
      then consider a2 such that
A5:   a2 in T and
A6:   S = InitSegm(T,a2) by A2,Def11;
      consider a1 such that
A7:   a1 in S and
A8:   T = InitSegm(S,a1) by A3,A4,Def11;
      a1 in LowerCone{a2} by A7,A6,XBOOLE_0:def 4;
      then
A9:   ex a st a = a1 & for b st b in {a2} holds a < b;
      a2 in LowerCone{a1} by A8,A5,XBOOLE_0:def 4;
      then
A10:  ex a3 st a3 = a2 & for b st b in {a1} holds a3 < b;
      a1 in {a1} by TARSKI:def 1;
      then
A11:  a2 < a1 by A10;
      a2 in {a2} by TARSKI:def 1;
      then a1 < a2 by A9;
      hence thesis by A11,Th4;
    end;
  end;
  hence thesis;
end;
