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;

theorem Th10:
  CQC_Subst(ll,Sub) is CQC-variable_list of k,Al
proof
  reconsider ll as FinSequence of bound_QC-variables(Al) by SUBSTUT1:34;
  reconsider s = CQC_Subst(ll,Sub) as FinSequence of bound_QC-variables(Al);
A1: s = CQC_Subst(@ll,Sub) by SUBSTUT1:def 5;
  len ll = k by CARD_1:def 7;
  then len @ll = k by SUBSTUT1:def 4;
  then len s = k by A1,SUBSTUT1:def 3;
  then s is CQC-variable_list of k,Al by SUBSTUT1:34;
  hence thesis by A1,SUBSTUT1:def 4;
end;
