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;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;

theorem Th42:
  fC1 c< fC2 iff fC1 is Initial_Segm of fC2
proof
  thus fC1 c< fC2 implies fC1 is Initial_Segm of fC2
  proof
    assume
A1: fC1 c< fC2;
    now
      assume
A2:   fC2 is Initial_Segm of fC1;
      fC1 <> {} by Def12;
      then ex a st a in fC1 & fC2 = InitSegm(fC1,a) by A2,Def11;
      then fC2 c= fC1 by XBOOLE_1:17;
      hence contradiction by A1,XBOOLE_0:def 8;
    end;
    hence thesis by A1,Th41;
  end;
  assume
A3: fC1 is Initial_Segm of fC2;
A4: fC2 <> {} by Def12;
  then ex a st a in fC2 & fC1 = InitSegm(fC2,a) by A3,Def11;
  then
A5: fC1 c= fC2 by XBOOLE_1:17;
  fC1 <> fC2 by A3,A4,Th30;
  hence thesis by A5;
end;
