
theorem
  for L being continuous LATTICE for p being Element of L st uparrow
fininfs (downarrow p)` misses waybelow p for A being finite non empty Subset of
  L st inf A << p ex a being Element of L st a in A & a <= p
proof
  let L be continuous LATTICE, p be Element of L such that
A1: uparrow fininfs (downarrow p)` misses waybelow p;
  let A be finite non empty Subset of L;
  assume inf A << p;
  then inf A in waybelow p;
  then not inf A in uparrow fininfs (downarrow p)` by A1,XBOOLE_0:3;
  then (downarrow p)` c= uparrow fininfs (downarrow p)` & not A c= uparrow
  fininfs (downarrow p)` by WAYBEL_0:43,62;
  then not A c= (downarrow p)`;
  then consider a being object such that
A2: a in A and
A3: not a in (downarrow p)`;
  reconsider a as Element of L by A2;
  take a;
  a in downarrow p by A3,SUBSET_1:29;
  hence thesis by A2,WAYBEL_0:17;
end;
