reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th41:
  for L being non empty reflexive transitive RelStr holds (x is
  Element of InclPoset(Ids L) iff x is Ideal of L)
proof
  let L be non empty reflexive transitive RelStr;
  hereby
    assume x is Element of InclPoset(Ids L);
    then x in the carrier of InclPoset(Ids L);
    then x in Ids L by YELLOW_1:1;
    then ex J being Ideal of L st J = x;
    hence x is Ideal of L;
  end;
  assume x is Ideal of L;
  then x in the set of all Y where Y is Ideal of L;
  hence thesis by YELLOW_1:1;
end;
