reserve A for QC-alphabet;
reserve k,n,m for Nat;
reserve P for QC-pred_symbol of A;

theorem Th8:
  for k being Nat, p being QC-pred_symbol of k, A, ll be
  QC-variable_list of k, A holds p!ll = <*p*>^ll
proof
  let k be Nat, p be QC-pred_symbol of k, A,
      ll be QC-variable_list of k, A;
  set QCP = {Q where Q is QC-pred_symbol of A: the_arity_of Q = 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,Def12;
end;
