
theorem
  for L be non empty reflexive transitive RelStr holds Ids L = Filt (L opp)
proof
  let L be non empty reflexive transitive RelStr;
A1: Filt (L opp) c= Ids L
  proof
    let x be object;
    assume x in Filt L opp;
    then x in the set of all  X where X is Filter of L opp  by
WAYBEL_0:def 24;
    then consider x1 be Filter of L opp such that
A2: x1 = x;
    x1 is lower Subset of L & x1 is directed Subset of L by YELLOW_7:26,28;
    then x in the set of all  X where X is Ideal of L  by A2;
    hence thesis by WAYBEL_0:def 23;
  end;
  Ids L c= Filt (L opp)
  proof
    let x be object;
    assume x in Ids L;
    then x in the set of all  X where X is Ideal of L  by
WAYBEL_0:def 23;
    then consider x1 be Ideal of L such that
A3: x1 = x;
    x1 is upper Subset of L opp & x1 is filtered Subset of L opp by YELLOW_7:26
,28;
    then x in the set of all  X where X is Filter of L opp  by A3;
    hence thesis by WAYBEL_0:def 24;
  end;
  hence thesis by A1;
end;
