
theorem
  for L being distributive complete LATTICE for x,y being Element of L
  holds x << y iff for P being prime Ideal of L st y <= sup P holds x in P
proof
  let L be distributive complete LATTICE;
  let x,y be Element of L;
  thus x << y implies for P being prime Ideal of L st y <= sup P holds x in P
  by WAYBEL_3:20;
  assume
A1: for P being prime Ideal of L st y <= sup P holds x in P;
  now
    let I be Ideal of L;
    assume that
A2: y <= sup I and
A3: not x in I;
    consider P being Ideal of L such that
A4: P is prime and
A5: I c= P and
A6: not x in P by A3,Th24;
    sup I <= sup P by A5,Th1;
    hence contradiction by A1,A2,A4,A6,ORDERS_2:3;
  end;
  hence thesis by WAYBEL_3:21;
end;
