
theorem
  for L being continuous LATTICE st Top L is compact holds (for A being
finite non empty Subset of L st inf A << Top L ex a being Element of L st a in
  A & a <= Top L) & uparrow fininfs (downarrow Top L)` meets waybelow Top L
proof
  let L be continuous LATTICE such that
A1: Top L << Top L;
A2: now
    take x = Top L;
    thus x in uparrow fininfs (downarrow Top L)` by WAYBEL_4:22;
    thus x in waybelow Top L by A1;
  end;
  hereby
    let A be finite non empty Subset of L such that
    inf A << Top L;
    set a = the Element of A;
    reconsider a as Element of L;
    take a;
    thus a in A & a <= Top L by YELLOW_0:45;
  end;
  thus thesis by A2,XBOOLE_0:3;
end;
