
theorem Th25:
  for A being Universal_Algebra, B,C being Subset of A
  st C is opers_closed & B c= C
  holds union the set of all B|^n where n is Element of NAT c= C
proof
  let A be Universal_Algebra;
  let B,C be Subset of A;
  assume
A1: C is opers_closed;
  assume
A2: B c= C;
  let z be object;
  assume z in union the set of all B|^n where n is Element of NAT;
  then consider Y being set such that
A3: z in Y and
A4: Y in the set of all B|^n where n is Element of NAT by TARSKI:def 4;
  consider n being Element of NAT such that
A5: Y = B|^n by A4;
  defpred P[Nat] means B|^$1 c= C;
A6: P[0] by A2,Th18;
A7: now
    let n be Nat;
    assume
A8: P[n];
    thus P[n+1]
    proof
      let x be object;
      assume that
A9:   x in B|^(n+1) and
A10:  x nin C;
      x in (B|^n)\/{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}
      by A9,Th19;
      then x in B|^n or
      x in {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}
      by XBOOLE_0:def 3;
      then consider o being (Element of dom the charact of A), p being Element
      of (the carrier of A)* such that
A11:  x = Den(o,A).p and
A12:  p in dom Den(o,A) and
A13:  rng p c= B|^n by A8,A10;
      rng p c= C by A8,A13,XBOOLE_1:1;
      then reconsider p as FinSequence of C by FINSEQ_1:def 4;
      reconsider oo = Den(o,A) as Element of Operations A;
A14:  len p = arity oo by A12,MARGREL1:def 25;
      C is_closed_on oo by A1;
      hence thesis by A10,A11,A14;
    end;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A6,A7);
  then P[n];
  hence thesis by A3,A5;
end;
