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 Th74:
  m <= n implies PrimRec-Approximation.m c= PrimRec-Approximation. n
proof
  set prd = PrimRec-Approximation;
  defpred p[Nat] means m <= $1 implies prd.m c= prd.$1;
A1: for n be Nat st p[n] holds p[n+1]
  proof
    let n be Nat such that
A2: p[n] and
A3: m <= n+1;
    prd.(n+1) = (PR-closure (prd.n)) \/ (composition-closure (prd.n)) by Def20;
    then
A4: PR-closure (prd.n) c= prd.(n+1) by XBOOLE_1:7;
    prd.n c= PR-closure (prd.n) by XBOOLE_1:7;
    then
A5: prd.n c= prd.(n+1) by A4;
    per cases by A3,XXREAL_0:1;
    suppose
      m < n+1;
      hence thesis by A2,A5,NAT_1:13;
    end;
    suppose
      m = n+1;
      hence thesis;
    end;
  end;
A6: p[0];
  for n being Nat holds p[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
