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 Th5:
  for k being Nat holds
    FCEx(k-th_FCEx(Al)) = (k+1)-th_FCEx(Al)
proof
  let k be Nat;
    k+1+1 <= k+1+1 & 0 < 0 + (k + 1);
  then k+1 < k+2 & 1 <= k+1 by NAT_1:19;
  then consider Al2 being QC-alphabet such that
A1: (the FCEx-Sequence of Al,k+1).(k+1) = Al2 &
    (the FCEx-Sequence of Al,k+1).(k+2) = FCEx(Al2) by Def7;
  set X = (the FCEx-Sequence of Al,k+1).(k+1);
  X = k-th_FCEx(Al)
  proof
    defpred A[Nat] means 0 < $1 & $1 <= k+1 implies
      (the FCEx-Sequence of Al,k).$1 = (the FCEx-Sequence of Al,k+1).$1;
A2: A[0];
A3: for n being Nat st A[n] holds A[n+1]
    proof
      let n be Nat;
      assume
A4:   A[n];
      per cases;
      suppose
A5:     (n+1) <= k+1;
        per cases;
        suppose
A6:       n=0;
          (the FCEx-Sequence of Al,k).1 = Al by Def7
           .= (the FCEx-Sequence of Al,k+1).1 by Def7;
          hence A[n+1] by A6;
        end;
        suppose
A7:       n <> 0;
A8:      0 < 0 + n & n <= n+1 by A7,NAT_1:11;
          0 < 0 + n by A7;
          then
A9:      1 <= n by NAT_1:19;
A10:      n < k+1 by A5,NAT_1:13;
          then n < k+1+1 by NAT_1:13;
          then
          consider Al3 being QC-alphabet such that
A11:       (the FCEx-Sequence of Al,k+1).n = Al3 &
           (the FCEx-Sequence of Al,k+1).(n+1) = FCEx(Al3) by A9,Def7;
          consider Al4 being QC-alphabet such that
A12:       (the FCEx-Sequence of Al,k).n = Al4 &
           (the FCEx-Sequence of Al,k).(n+1) = FCEx(Al4) by A9,A10,Def7;
          thus A[n+1] by A4,A8,A11,A12,XXREAL_0:2;
        end;
      end;
      suppose not n+1 <= k+1;
        hence A[n+1];
      end;
    end;
    for n being Nat holds A[n] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
  hence thesis by A1;
end;
