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

theorem Th42:
  for L being non empty reflexive transitive RelStr for I being
  Element of InclPoset(Ids L) holds x in I implies x is Element of L
proof
  let L be non empty reflexive transitive RelStr;
  let I be Element of InclPoset(Ids L);
  reconsider I9= I as non empty Subset of L by Th41;
  assume x in I;
  then reconsider x9= x as Element of I9;
  x9 in the carrier of L;
  hence thesis;
end;
