reserve Al for QC-alphabet;
reserve a,a1,a2,b,c,d for set,
  X,Y,Z for Subset of CQC-WFF(Al),
  i,k,m,n for Nat,
  p,q for Element of CQC-WFF(Al),
  P for QC-pred_symbol of k,Al,
  ll for CQC-variable_list of k,Al,
  f,f1,f2,g for FinSequence of CQC-WFF(Al);
reserve A for non empty finite Subset of NAT;

theorem Th2:
  union A in A & for a st a in A holds (a in union A or a = union A )
proof
  consider a being Ordinal such that
A1: RelIncl A,RelIncl a are_isomorphic by WELLORD2:8,13;
  consider F1 being Function such that
A2: F1 is_isomorphism_of RelIncl A,RelIncl a by A1,WELLORD1:def 8;
A3: dom F1 = field RelIncl A by A2,WELLORD1:def 7;
  then dom F1 = A by WELLORD2:def 1;
  then consider b being object such that
A4: b in dom F1 by XBOOLE_0:def 1;
  rng F1 is finite by A3,FINSET_1:8;
  then field RelIncl a is finite by A2,WELLORD1:def 7;
  then
A5: a is finite by WELLORD2:def 1;
  F1.b in rng F1 by A4,FUNCT_1:3;
  then field RelIncl a is non empty by A2,WELLORD1:def 7;
  then a is non empty by WELLORD2:def 1;
  then
A6: {} in a by ORDINAL1:16,XBOOLE_1:3;
A7: now
    assume a is limit_ordinal;
    then omega c= a by A6,ORDINAL1:def 11;
    hence contradiction by A5;
  end;
  RelIncl a,RelIncl A are_isomorphic by A1,WELLORD1:40;
  then consider F being Function such that
A8: F is_isomorphism_of RelIncl a,RelIncl A by WELLORD1:def 8;
A9: for m,n st m in dom F & n in dom F & n < m holds F.n in F.m
  proof
    let m,n such that
A10: m in dom F and
A11: n in dom F and
A12: n < m;
    F.n in rng F by A11,FUNCT_1:3;
    then F.n in field RelIncl A by A8,WELLORD1:def 7;
    then
A13: F.n in A by WELLORD2:def 1;
    then reconsider b = F.n as Nat;
    n in field RelIncl a by A8,A11,WELLORD1:def 7;
    then
A14: n in a by WELLORD2:def 1;
    F.m in rng F by A10,FUNCT_1:3;
    then F.m in field RelIncl A by A8,WELLORD1:def 7;
    then
A15: F.m in A by WELLORD2:def 1;
    then reconsider c = F.m as Nat;
    n in {i where i is Nat: i < m } by A12;
    then n in m by AXIOMS:4;
    then
A16: n c= m by ORDINAL1:def 2;
    m in field RelIncl a by A8,A10,WELLORD1:def 7;
    then m in a by WELLORD2:def 1;
    then [n,m] in RelIncl a by A14,A16,WELLORD2:def 1;
    then [F.n,F.m] in RelIncl A by A8,WELLORD1:def 7;
    then
A17: F.n c= F.m by A13,A15,WELLORD2:def 1;
    F is one-to-one by A8,WELLORD1:def 7;
    then F.n <> F.m by A10,A11,A12;
    then F.n c< F.m by A17;
    then b in c by ORDINAL1:11;
    hence thesis;
  end;
  reconsider a as Nat by A5;
  dom F = field RelIncl a by A8,WELLORD1:def 7;
  then
A18: dom F = a by WELLORD2:def 1;
A19: now
    let b be Ordinal;
A20: a in NAT by ORDINAL1:def 12;
    assume succ b = a;
    then b in a by ORDINAL1:6;
    hence b in NAT by ORDINAL1:10,A20;
  end;
A21: ex m st succ m = a
  proof
    consider b being Ordinal such that
A22: succ b = a by A7,ORDINAL1:29;
    reconsider b as Element of NAT by A19,A22;
    take b;
    thus thesis by A22;
  end;
  then consider m such that
A23: succ m = a;
A24: rng F = field RelIncl A by A8,WELLORD1:def 7;
  then
A25: rng F = A by WELLORD2:def 1;
  then reconsider F as Function of a,A by A18,FUNCT_2:1;
A26: for n,m st m in dom F & n in dom F & n < m holds F.n in F.m by A9;
  rng F = A by A24,WELLORD2:def 1;
  then
A27: F.(union a) = union rng F by A21,A26,Th1;
A28: union a = m by A23,ORDINAL2:2;
  hence union A in A by A25,A18,A23,A27,FUNCT_1:3,ORDINAL1:6;
  thus for b st b in A holds (b in union A or b = union A)
  proof
    let b such that
A29: b in A;
    now
A30:  m in dom F by A18,A23,ORDINAL1:6;
      assume
A31:  b <> union A;
      consider c being object such that
A32:  c in dom F and
A33:  F.c = b by A25,A29,FUNCT_1:def 3;
    dom F = a by PARTFUN1:def 2;
    then
A34: dom F in NAT by ORDINAL1:def 12;
      reconsider c as Element of NAT by A32,ORDINAL1:10,A34;
      c in m or c = m by A23,A32,ORDINAL1:8;
      then c in {k where k is Nat : k < m}
       by A25,A23,A27,A31,A33,AXIOMS:4,ORDINAL2:2;
      then ex k being Nat st k = c & k < m;
      hence thesis by A9,A25,A27,A28,A32,A33,A30;
    end;
    hence thesis;
  end;
end;
