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 Th4:
  for k being Nat for S being FCEx-Sequence of Al,k holds
   S.(k+1) is Al-expanding QC-alphabet
proof
  let k be Nat;
  let S be FCEx-Sequence of Al,k;
  set Al2 = S.(k+1);
  reconsider Al2 as QC-alphabet by Th3;
  Al c= Al2
  proof
    let x be object;
    assume
A1: x in Al;
    defpred A[Nat] means $1 < k+1 implies x in S.($1+1);
A2: A[0] by A1,Def7;
A3: for l being Nat st A[l] holds A[l+1]
    proof
      let l be Nat;
      assume
A4:   A[l];
      assume
A5:   l+1 < k+1;
A6:   for n being Nat st n+1 < (k+1) & 1 <= n+1 holds
      ex Al2 being QC-alphabet st S.(n+1) = Al2 & S.(n+2) = FCEx(Al2)
      proof
        let n be Nat such that
A7:     n + 1 < k + 1 & 1 <= n+1;
        set m = n +1;
        ex Al2 being QC-alphabet st S.(m) = Al2 & S.(m+1) = FCEx(Al2)
         by Def7,A7;
        hence thesis;
      end;
      0 < 0 + (l + 1);
      then consider Al2 being QC-alphabet such that
A8:   S.(l+1) = Al2 & S.(l+2) = FCEx(Al2) by A5,A6,NAT_1:19;
      S.(l+1) c= S.(l+2) by A8,QC_TRANS:def 1;
      hence thesis by A4,A5,NAT_1:16,XXREAL_0:2;
    end;
    for n being Nat holds A[n] from NAT_1:sch 2(A2,A3);
    then k < k+1 implies x in S.(k+1);
    hence thesis by NAT_1:13;
  end;
  hence thesis by QC_TRANS:def 1;
end;
