reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem Th21:
  x.t in still_not-bound_in p implies t < index p
proof
  assume
A1: x.t in still_not-bound_in p;
  now
    min NBI p in NBI p by QC_LANG1:def 35;
    then
A2: ex u st u = min NBI p & for t st u<=t holds not x.t in
    still_not-bound_in p;
    assume min (NBI p) <= t;
    hence contradiction by A1,A2;
  end;
  hence thesis by QC_LANG1:25;
end;
