
theorem Th16:
  for L1 be non empty finite reflexive transitive RelStr holds Ids L1 is finite
proof
  deffunc F(set) = $1;
  let L1 be non empty finite reflexive transitive RelStr;
  reconsider Y = bool the carrier of L1 as finite non empty set;
A1: the set of all  X where X is Ideal of L1  c= { X where X is
  Element of Y : X is Ideal of L1 }
  proof
    let z be object;
    assume z in the set of all  X where X is Ideal of L1 ;
    then ex X1 be Ideal of L1 st z = X1;
    hence thesis;
  end;
  defpred P[set] means $1 is Ideal of L1;
A2: { X where X is Element of Y : X is Ideal of L1 } c= the set of all
 X where X is Ideal
  of L1
  proof
    let z be object;
    assume z in { X where X is Element of Y : X is Ideal of L1 };
    then ex X1 be Element of Y st z = X1 & X1 is Ideal of L1;
    hence thesis;
  end;
A3: {F(X) where X is Element of Y : P[X]} is finite from PRE_CIRC:sch 1;
  Ids L1 = the set of all  X where X is Ideal of L1  by WAYBEL_0:def 23
    .= { X where X is Element of Y : X is Ideal of L1 } by A1,A2,
XBOOLE_0:def 10;
  hence thesis by A3;
end;
