
theorem
  for L being continuous LATTICE for p being Element of L st (p <> Top L
or Top L is not compact) & 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 holds uparrow fininfs (downarrow p
  )` misses waybelow p
proof
  let L be continuous LATTICE, p be Element of L such that
A1: p <> Top L or Top L is not compact and
A2: 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;
  now
    let x be object;
    assume
A3: x in uparrow fininfs (downarrow p)`;
    then reconsider a = x as Element of L;
    consider b being Element of L such that
A4: a >= b and
A5: b in fininfs (downarrow p)` by A3,WAYBEL_0:def 16;
    assume x in waybelow p;
    then
A6: a << p by WAYBEL_3:7;
    then
A7: b << p by A4,WAYBEL_3:2;
    consider Y being finite Subset of (downarrow p)` such that
A8: b = "/\"(Y,L) and
    ex_inf_of Y,L by A5;
    reconsider Y as finite Subset of L by XBOOLE_1:1;
    per cases;
    suppose
      Y <> {};
      then consider c being Element of L such that
A9:   c in Y and
A10:  c <= p by A2,A4,A8,A6,WAYBEL_3:2;
      c in downarrow p by A10,WAYBEL_0:17;
      then (downarrow p) misses (downarrow p)` & c in (downarrow p)/\(
      downarrow p)` by A9,XBOOLE_0:def 4,XBOOLE_1:79;
      hence contradiction;
    end;
    suppose
A11:  Y = {};
      b <= p & p <= Top L by A7,WAYBEL_3:1,YELLOW_0:45;
      then p = Top L by A8,A11,ORDERS_2:2;
      hence contradiction by A1,A8,A7,A11;
    end;
  end;
  hence thesis by XBOOLE_0:3;
end;
