reserve Al for QC-alphabet,
     PHI for Consistent Subset of CQC-WFF(Al),
     PSI for Subset of CQC-WFF(Al),
     p,q,r,s for Element of CQC-WFF(Al),
     A for non empty set,
     J for interpretation of Al,A,
     v for Element of Valuations_in(Al,A),
     m,n,i,j,k for Element of NAT,
     l for CQC-variable_list of k,Al,
     P for QC-pred_symbol of k,Al,
     x,y for bound_QC-variable of Al,
     z for QC-symbol of Al,
     Al2 for Al-expanding QC-alphabet;
reserve J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th2:
 for k being Element of NAT st k > 0 ex F being k-element FinSequence st
 (for n being Nat st n <= k & 1 <= n holds F.n is QC-alphabet) &
 F.1 = Al &
 for n being Nat st n < k & 1 <= n holds
    ex Al2 being QC-alphabet st F.n = Al2 & F.(n+1) = FCEx(Al2)
proof
  defpred A[Nat] means $1 > 0 implies
   ex F being $1-element FinSequence st
   ( for n being Nat st n <= $1 & 1 <= n holds F.n is QC-alphabet ) &
    F.1 = Al & ( for n being Nat st n < $1 & 1 <= n holds
    ex Al2 being QC-alphabet st F.n = Al2 & F.(n+1) = FCEx(Al2));
A1: for k being Nat st A[k] holds A[k+1]
  proof
    let k be Nat;
    assume
A2: A[k];
    per cases;
    suppose
A3:   k = 0;
A4:   <*Al*> is 1-element FinSequence & <*Al*>.1 = Al;
A5:   for n being Nat st
      n < 1 & 1 <= n holds ex Al2 being QC-alphabet
      st <*Al*>.n = Al2 & <*Al*>.(n+1) = FCEx(Al2);
      for n being Nat st n<=1 & 1<=n holds <*Al*>.n is QC-alphabet
       by A4,XXREAL_0:1;
      hence thesis by A3,A4,A5;
    end;
    suppose
A6:   k <> 0;
      then consider F being k-element FinSequence such that
A7:    ( for n being Nat st n <= k & 1 <= n holds F.n is QC-alphabet ) &
       ( F.1  = Al ) & ( for n being Nat st n < k & 1 <= n holds ex Al2 being
       QC-alphabet st F.n = Al2 & F.(n+1) = FCEx(Al2)) by A2;
      set K = F.k;
      K is QC-alphabet
      proof
        per cases;
        suppose k = 1;
          hence thesis by A7;
        end;
        suppose
A8:       k <> 1;
          consider j being Nat such that
A9:        k = j+1 by NAT_1:6, A6;
          j <> 0 by A8,A9;
          then j >= 1 & j < k by A9,NAT_1:25,19;
          then ex Al2 being QC-alphabet st F.j = Al2 & F.(k) = FCEx(Al2)
            by A7,A9;
          hence thesis;
        end;
      end;
      then reconsider K as QC-alphabet;
      set K2 = <*FCEx(K)*>;
      set F2 = F^K2;
      reconsider F2 as (k+1)-element FinSequence;
A10:  1 <= k & len F = k by A6,NAT_1:25,CARD_1:def 7;
A11:  for n being Nat st n < k & 1 <= n holds
       ex Al2 being QC-alphabet st F2.n = Al2 & F2.(n+1) = FCEx(Al2)
      proof
        let n be Nat such that
A12:      n < k & 1 <= n;
        consider Al2 being QC-alphabet such that
A13:     F.n = Al2 & F.(n+1) = FCEx(Al2) by A7,A12;
        1 <= n+1 & n+1 <= k & k = len F by A12,NAT_1:13,CARD_1:def 7;
        then F2.n = F.n & F2.(n+1) = F.(n+1) by A12,FINSEQ_1:64;
        hence thesis by A13;
      end;
A14:  K is QC-alphabet & F2.k = K by A10,FINSEQ_1:64;
A15:  for n being Nat st n < k+1 & 1 <= n holds
       ex Al2 being QC-alphabet st F2.n = Al2 & F2.(n+1) = FCEx(Al2)
      proof
        let n be Nat such that
A16:      n < k+1 & 1 <= n;
        per cases;
        suppose n <> k;
          hence thesis by A11,A16,NAT_1:22;
        end;
        suppose n = k;
          hence thesis by A10,A14,FINSEQ_1:42;
        end;
      end;
A17:  for n being Nat st n <= k+1 & 1 <= n holds F2.n is QC-alphabet
      proof
        let n be Nat such that
A18:      n <= k+1 & 1 <= n;
        per cases;
        suppose n <> k+1;
          then n <= k by A18,NAT_1:8;
          then F2.n = F.n & F.n is QC-alphabet by A7,A10,A18,FINSEQ_1:64;
          hence thesis;
        end;
        suppose n = k +1;
          hence thesis by A10,FINSEQ_1:42;
        end;
      end;
      F2.1 = Al by A7,A10,FINSEQ_1:64;
      hence thesis by A15,A17;
    end;
  end;
A19: A[0];
  for n being Nat holds A[n] from NAT_1:sch 2(A19,A1);
  hence thesis;
end;
