
theorem Th36:
  for L being lower-bounded continuous LATTICE, x being Element of
L for N being non empty countable Subset of L st for n, y being Element of L st
  not y <= x & n in N holds not y "/\" n <= x for y being Element of L st not y
<= x ex p being irreducible Element of L st x <= p & not p in uparrow ({y} "/\"
  N)
proof
  let L be lower-bounded continuous LATTICE, x be Element of L, N be non empty
  countable Subset of L such that
A1: for n, y being Element of L st not y <= x & n in N holds not y "/\" n <= x;
  set V = (downarrow x)`;
A2: V "/\" N c= V
  proof
    let q be object;
    assume q in V "/\" N;
    then consider v, n being Element of L such that
A3: q = v "/\" n and
A4: v in V and
A5: n in N;
    not v in downarrow x by A4,XBOOLE_0:def 5;
    then not v <= x by WAYBEL_0:17;
    then not v "/\" n <= x by A1,A5;
    then not v "/\" n in downarrow x by WAYBEL_0:17;
    hence thesis by A3,XBOOLE_0:def 5;
  end;
  x <= x;
  then x in downarrow x by WAYBEL_0:17;
  then
A6: not x in V by XBOOLE_0:def 5;
  let y be Element of L;
  assume not y <= x;
  then not y in downarrow x by WAYBEL_0:17;
  then y in V by XBOOLE_0:def 5;
  hence thesis by A2,A6,Th35;
end;
