reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th14: ::3.7, p.69
  for L be lower-bounded continuous LATTICE, x,y be Element of L
  st not y <= x holds ex p be Element of L st p is irreducible & x <= p & not y
  <= p
proof
  let L be lower-bounded continuous LATTICE, x,y be Element of L such that
A1: not y <= x;
  for x being Element of L holds waybelow x is non empty directed;
  then consider u being Element of L such that
A2: u << y and
A3: not u <= x by A1,WAYBEL_3:24;
  consider F be Open Filter of L such that
A4: y in F and
A5: F c= wayabove u by A2,Th8;
A6: wayabove u c= uparrow u by WAYBEL_3:11;
  not x in F by A3,WAYBEL_0:18,A5,A6;
  then x in (the carrier of L)\F by XBOOLE_0:def 5;
  then consider m being Element of L such that
A7: x <= m and
A8: m is_maximal_in (F`) by Th9;
  take m;
A9: m in (F`) by A8,WAYBEL_4:55;
  now
    assume y <= m;
    then m in F by A4,WAYBEL_0:def 20;
    hence contradiction by A9,XBOOLE_0:def 5;
  end;
  hence thesis by A7,A8,Th13;
end;
