reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;
reserve f1, f2 for non empty homogeneous to-naturals NAT*-defined Function,
  e1, e2 for homogeneous to-naturals NAT*-defined Function,
  p for Element of (arity f1+1)-tuples_on NAT;
reserve P for primitive-recursively_closed non empty Subset of HFuncs NAT;

theorem Th73:
  for X being Subset of HFuncs(NAT) st X is primitive-recursively_closed
  holds PrimRec c= X
proof
  let X be Subset of HFuncs(NAT);
  set S = { R where R is Subset of HFuncs(NAT) : R is
  primitive-recursively_closed };
  assume X is primitive-recursively_closed;
  then
A1: X in S;
  let x be object;
  assume x in PrimRec;
  hence thesis by A1,SETFAM_1:def 1;
end;
