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 Th6:
 for k,n st n <= k holds n-th_FCEx(Al) c= k-th_FCEx(Al)
proof
  let k;
  defpred P[Nat] means $1 <= k implies ex j st j = k-$1 &
   j-th_FCEx(Al) c= k-th_FCEx(Al);
A1: P[0];
A2: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat such that
A3:  P[n];
    per cases;
    suppose
A4:   n+1 <= k;
      then consider j such that
A5:    j = k-n & j-th_FCEx(Al) c= k-th_FCEx(Al) by A3,NAT_1:13;
      set j2=k-(n+1);
      reconsider j2 as Element of NAT by A4,NAT_1:21;
      FCEx(j2-th_FCEx(Al)) = j-th_FCEx(Al) by A5,Th5;
      then j2-th_FCEx(Al) c= j-th_FCEx(Al) by QC_TRANS:def 1;
      hence thesis by A5,XBOOLE_1:1;
    end;
    suppose not n+1 <= k;
      hence thesis;
    end;
  end;
A6: for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
  let n such that
A7: n <= k;
  set n2 = k - n;
  reconsider n2 as Element of NAT by A7,NAT_1:21;
  k = n + n2;
  then consider n3 being Element of NAT such that
A8: n3 = k-n2 & n3-th_FCEx(Al) c= k-th_FCEx(Al) by A6,NAT_1:11;
  thus thesis by A8;
end;
