reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

theorem Th59:
  v*'ll = ll*(v|still_not-bound_in ll)
proof
  rng ll c= bound_QC-variables(Al) by RELAT_1:def 19;
  then
A1: rng ll = still_not-bound_in ll by Th57;
  dom (v|still_not-bound_in ll) = dom v /\ still_not-bound_in ll by RELAT_1:61;
  then
  dom (v|still_not-bound_in ll) = bound_QC-variables(Al) /\ still_not-bound_in
  ll by Th58;
  then rng ll = dom (v|still_not-bound_in ll) by A1,XBOOLE_1:28;
  then
A2: dom (ll*(v|still_not-bound_in ll)) = dom ll by RELAT_1:27;
  then
A3: dom (ll*(v|still_not-bound_in ll)) = Seg len ll by FINSEQ_1:def 3;
  then reconsider f = ll*(v|still_not-bound_in ll) as FinSequence by
FINSEQ_1:def 2;
  len f = len ll by A3,FINSEQ_1:def 3;
  then
A4: len f = k by SUBSTUT1:34;
  then
A5: dom f = Seg k by FINSEQ_1:def 3;
A6: for j be Nat st j in dom f holds f.j = (v*'ll).j
  proof
A7: rng ll c= bound_QC-variables(Al) by RELAT_1:def 19;
    let j be Nat such that
A8: j in dom f;
    reconsider j as Nat;
    ll.j in rng ll by A2,A8,FUNCT_1:3;
    then ll.j in bound_QC-variables(Al) by A7;
    then
A9: ll.j in dom v by Th58;
    ll.j in still_not-bound_in ll by A1,A2,A8,FUNCT_1:3;
    then ll.j in dom v /\ still_not-bound_in ll by A9,XBOOLE_0:def 4;
    then
A10: (v|still_not-bound_in ll).(ll.j) = v.(ll.j) by FUNCT_1:48;
    1 <= j & j <= k by A5,A8,FINSEQ_1:1;
    then (v|still_not-bound_in ll).(ll.j) = (v*'ll).j by A10,VALUAT_1:def 3;
    hence thesis by A2,A8,FUNCT_1:13;
  end;
  len (v*'ll) = k by VALUAT_1:def 3;
  hence thesis by A4,A6,FINSEQ_2:9;
end;
