reserve Al for QC-alphabet;
reserve a,a1,a2,b,c,d for set,
  X,Y,Z for Subset of CQC-WFF(Al),
  i,k,m,n for Nat,
  p,q for Element of CQC-WFF(Al),
  P for QC-pred_symbol of k,Al,
  ll for CQC-variable_list of k,Al,
  f,f1,f2,g for FinSequence of CQC-WFF(Al);
reserve A for non empty finite Subset of NAT;
reserve C for non empty set;
reserve A for non empty set,
  v for Element of Valuations_in(Al,A),
  J for interpretation of Al,A;
reserve CX for Consistent Subset of CQC-WFF(Al),
  P9 for Element of QC-pred_symbols(Al);
reserve JH for Henkin_interpretation of CX;

theorem Th14:
  (valH(Al))*'ll = ll
proof
A1: for i st i in dom ((valH(Al))*'ll) holds (valH(Al)).(ll.i) = ll.i
  proof
A2: dom((valH(Al))*'ll) c= dom ll by RELAT_1:25;
    let i;
    assume i in dom ((valH(Al))*'ll);
    then
A3: ll.i in rng ll by A2,FUNCT_1:3;
    rng ll c= bound_QC-variables(Al) by RELAT_1:def 19;
    hence thesis by A3,FUNCT_1:18;
  end;
A4: for i st 1 <= i & i <= k holds (valH(Al)).(ll.i) = ll.i
  proof
    let i such that
A5: 1 <= i and
A6: i <= k;
    i <= len ((valH(Al))*'ll) by A6,VALUAT_1:def 3;
    then i in dom ((valH(Al))*'ll) by A5,FINSEQ_3:25;
    hence thesis by A1;
  end;
A7: len ll = k by CARD_1:def 7;
  then
A8: dom ll = Seg k by FINSEQ_1:def 3;
A9: for i being Nat st i in dom ll holds ((valH(Al))*'ll).i = ll.i
  proof
    let i be Nat such that
A10: i in dom ll;
    reconsider i as Nat;
A11: 1 <= i & i <= k by A8,A10,FINSEQ_1:1;
    then ((valH(Al))*'ll).i = (valH(Al)).(ll.i) by VALUAT_1:def 3;
    hence thesis by A4,A11;
  end;
  len ((valH(Al))*'ll) = k by VALUAT_1:def 3;
  hence thesis by A7,A9,FINSEQ_2:9;
end;
