
theorem Th54:
  for L be non empty reflexive transitive RelStr for S be non
  empty full SubRelStr of L for x be set holds x in dom idsMap S iff x is Ideal
  of S
proof
  let L be non empty reflexive transitive RelStr;
  let S be non empty full SubRelStr of L;
  let x be set;
  hereby
    assume x in dom idsMap S;
    then x in Ids S by Th53;
    then x in the set of all  X where X is Ideal of S  by
WAYBEL_0:def 23;
    then ex I be Ideal of S st x = I;
    hence x is Ideal of S;
  end;
  assume x is Ideal of S;
  then x in the set of all  X where X is Ideal of S;
  then x in Ids S by WAYBEL_0:def 23;
  hence thesis by Th53;
end;
