
theorem Th20:
  for A being Universal_Algebra, B being Subset of A for n being Nat
  for x being set holds x in B|^(n+1) iff x in B|^n or
  ex o being Element of dom the charact of A st
  ex p being Element of (the carrier of A)*
  st x = Den(o,A).p & p in dom Den(o,A) & rng p c= B|^n
proof
  let A be Universal_Algebra;
  let B be Subset of A;
  let n be Nat;
  set Z = {Den(o,A).p where o is (Element of dom the charact of A),
  p is Element of (the carrier of A)*: p in dom Den(o,A) & rng p c= B|^n};
  let x be set;
  B|^(n+1) = (B|^n) \/ Z by Th19;
  then x in B|^(n+1) iff x in B|^n or x in Z by XBOOLE_0:def 3;
  hence thesis;
end;
