theorem Th11:
  for k being Nat, P being QC-pred_symbol of k, A holds
  the_arity_of P = k
proof
  let k be Nat, P be QC-pred_symbol of k, A;
  reconsider P as Element of k-ary_QC-pred_symbols(A);
  P in { Q : the_arity_of Q = k };
  then ex Q st P = Q & the_arity_of Q = k;
  hence thesis;
end;
