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;
reserve C for non empty set;

theorem
  for f being sequence of bool CQC-WFF(Al) st (for n,m st m in dom f & n
  in dom f & n < m holds f.n is Consistent & f.n c= f.m) holds (union rng f) is
  Consistent
proof
  let f be sequence of bool CQC-WFF(Al);
  assume
A1: for n,m st m in dom f & n in dom f & n < m holds f.n is Consistent &
  f.n c= f.m;
  now
A2: for n st n in dom f holds f.n is Consistent
    proof
      let n such that
A3:   n in dom f;
      n+1 in NAT;
      then n < n+1 & n+1 in dom f by FUNCT_2:def 1,NAT_1:13;
      hence thesis by A1,A3;
    end;
    assume not union rng f is Consistent;
    then consider Z such that
A4: Z c= union rng f & Z is finite and
A5: Z is Inconsistent by Th7;
    for n,m st m in dom f & n in dom f & n < m holds f.n c= f.m by A1;
    then consider k such that
A6: Z c= f.k by A4,Th3;
    reconsider Y = f.k as Subset of CQC-WFF(Al);
    consider p such that
A7: Z |- p and
A8: Z |- 'not' p by A5;
    consider f1 such that
A9: rng f1 c= Z and
A10: |- f1^<*p*> by A7;
    consider f2 such that
A11: rng f2 c= Z and
A12: |- f2^<*'not' p*> by A8;
    rng(f1^f2) = rng f1 \/ rng f2 by FINSEQ_1:31;
    then rng(f1^f2) c= Z by A9,A11,XBOOLE_1:8;
    then
A13: rng(f1^f2) c= Y by A6;
    |- f1^f2^<*'not' p*> by A12,CALCUL_2:20;
    then
A14: Y |- 'not' p by A13;
    |- f1^f2^<*p*> by A10,Th5;
    then Y |- p by A13;
    then
A15: not Y is Consistent by A14;
    k in NAT by ORDINAL1:def 12;
    then k in dom f by FUNCT_2:def 1;
    hence contradiction by A15,A2;
  end;
  hence thesis;
end;
