reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;
reserve Z for Element of [:QC-WFF(A),vSUB(A):];

theorem Th34:
  for h being FinSequence holds h is CQC-variable_list of k,A iff h
  is FinSequence of bound_QC-variables(A) & len h = k
proof
  let h be FinSequence;
  thus h is CQC-variable_list of k,A implies h is FinSequence of
  bound_QC-variables(A) & len h = k
  proof
    assume
A1: h is CQC-variable_list of k,A;
    then rng h c= bound_QC-variables(A) by RELAT_1:def 19;
    hence h is FinSequence of bound_QC-variables(A) by FINSEQ_1:def 4;
    thus thesis by A1,CARD_1:def 7;
  end;
  thus h is FinSequence of bound_QC-variables(A) & len h = k implies h is
  CQC-variable_list of k,A
  proof
    assume that
A2: h is FinSequence of bound_QC-variables(A) and
A3: len h = k;
    rng h c= bound_QC-variables(A) by A2,FINSEQ_1:def 4;
    then rng h c= QC-variables(A) by XBOOLE_1:1;
    hence thesis by A2,A3,CARD_1:def 7,FINSEQ_1:def 4;
  end;
end;
