
theorem Th55:
  for L be non empty reflexive transitive RelStr for S be non
  empty full SubRelStr of L for x be set holds x in rng idsMap S implies x is
  Ideal of L
proof
  let L be non empty reflexive transitive RelStr;
  let S be non empty full SubRelStr of L;
  let x be set;
  assume
A1: x in rng idsMap S;
  rng idsMap S is Subset of Ids L by Th53;
  then x in Ids L by A1;
  then x in the set of all  X where X is Ideal of L  by WAYBEL_0:def 23;
  then ex I be Ideal of L st x = I;
  hence thesis;
end;
