reserve x,y,X,X1,Y,Z for set;

theorem
  for X st X <> {} & for Z st Z <> {} & Z c= X & Z is c=-linear
  ex Y st Y in X & for X1 st X1 in Z holds X1 c= Y
  ex Y st Y in X & for Z st Z in X & Z <> Y holds not Y c= Z
proof
  let X such that
A1: X <> {} and
A2: for Z st Z <> {} & Z c= X & Z is c=-linear ex Y st Y in X & for X1
  st X1 in Z holds X1 c= Y;
  for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z
  holds X1 c= Y
  proof
    let Z such that
A3: Z c= X & Z is c=-linear;
    per cases;
    suppose
A4:   Z = {};
      set Y = the Element of X;
      for X1 st X1 in Z holds X1 c= Y by A4;
      hence thesis by A1;
    end;
    suppose
      Z <> {};
      hence thesis by A2,A3;
    end;
  end;
  hence thesis by A1,ORDERS_1:65;
end;
