
theorem
  for L being non empty reflexive transitive RelStr, F being Filter of L holds
  F is principal iff ex x being Element of L st F = uparrow x
proof
  let L be non empty reflexive transitive RelStr, I be Filter of L;
  thus I is principal implies ex x being Element of L st I = uparrow x
  proof
    given x being Element of L such that
A1: x in I and
A2: x is_<=_than I;
    take x;
    thus I c= uparrow x
    by A2,Th18;
    let z be object;
    assume
A3: z in uparrow x;
    then reconsider z as Element of L;
    z >= x by A3,Th18;
    hence thesis by A1,Def20;
  end;
  given x being Element of L such that
A4: I = uparrow x;
  take x;
  x <= x;
  hence x in I by A4,Th18;
  let y be Element of L;
  thus thesis by A4,Th18;
end;
