
theorem Th35:
  for L being lower-bounded continuous LATTICE, V being Open upper
Subset of L for N being non empty countable Subset of L, x, y being Element of
L st V "/\" N c= V & y in V & not x in V ex p being irreducible Element of L st
  x <= p & not p in uparrow ({y} "/\" N)
proof
  let L be lower-bounded continuous LATTICE, V be Open upper Subset of L, N be
  non empty countable Subset of L, x, y be Element of L such that
A1: V "/\" N c= V & y in V and
A2: not x in V;
  consider O being Open Filter of L such that
A3: {y} "/\" N c= O and
A4: O c= V and
  y in O by A1,Th34;
  uparrow O c= O & O c= uparrow O by WAYBEL_0:16,24;
  then uparrow O = O;
  then
A5: uparrow({y} "/\" N) c= O by A3,YELLOW_3:7;
  not x in O by A2,A4;
  then x in O` by XBOOLE_0:def 5;
  then consider p being Element of L such that
A6: x <= p and
A7: p is_maximal_in O` by WAYBEL_6:9;
  reconsider p as irreducible Element of L by A7,WAYBEL_6:13;
  take p;
  thus x <= p by A6;
  p in O` by A7,WAYBEL_4:55;
  hence thesis by A5,XBOOLE_0:def 5;
end;
