
theorem
  for L being non empty reflexive transitive RelStr, I being Ideal of L holds
  I is principal iff ex x being Element of L st I = downarrow x
proof
  let L be non empty reflexive transitive RelStr, I be Ideal of L;
  thus I is principal implies ex x being Element of L st I = downarrow 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= downarrow x
    by A2,Th17;
    let z be object;
    assume
A3: z in downarrow x;
    then reconsider z as Element of L;
    z <= x by A3,Th17;
    hence thesis by A1,Def19;
  end;
  given x being Element of L such that
A4: I = downarrow x;
  take x;
  x <= x;
  hence x in I by A4,Th17;
  let y be Element of L;
  thus thesis by A4,Th17;
end;
