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);

theorem Th9:
  for k being Nat, P being QC-pred_symbol of k,A, ll being
  QC-variable_list of k,A holds Sub_P(P,ll,e) = [P!ll,e]
proof
  let k be Nat, P be QC-pred_symbol of k,A,
ll be QC-variable_list of k, A;
  set QCP = {QP where QP is QC-pred_symbol of A: the_arity_of QP = k };
  P in QCP;
  then
A1: ex Q being QC-pred_symbol of A st P = Q & the_arity_of Q = k;
  len ll = k by CARD_1:def 7;
  hence thesis by A1,Def18;
end;
