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 Th69:
  1 <= i & i <= n implies n succ i in P
proof
A1: 1 succ 1 in P by Def14;
A2: arity (1 succ 1) = len <*n proj i*> by FINSEQ_1:39;
  reconsider nproji = n proj i as Element of HFuncs NAT by Th27;
  assume that
A3: 1 <= i and
A4: i <= n;
A5: <*nproji*> is with_the_same_arity FinSequence of HFuncs NAT;
  now
    rng (n proj i) = NAT by A3,A4,Th35;
    then
A6: rng <:<*n proj i*>:> = 1-tuples_on NAT by Th8;
    thus dom (n succ i) = n-tuples_on NAT by Def7;
A7: dom (n proj i) = n-tuples_on NAT by Th35;
    then
A8: dom <:<*n proj i*>:> = n-tuples_on NAT by FINSEQ_3:141;
    dom (1 succ 1) = 1-tuples_on NAT by Def7;
    hence
    dom ((1 succ 1)*<:<*n proj i*>:>)=n-tuples_on NAT by A8,A6,RELAT_1:27;
    let x be object;
    assume x in n-tuples_on NAT;
    then reconsider x9 = x as Element of n-tuples_on NAT;
    set xi = x9.i;
A10: (n succ i).x = xi+1 by Def7;
    reconsider ii= <*xi*> as Element of 1-tuples_on NAT by FINSEQ_2:131;
    ((1 succ 1)*<:<*n proj i*>:>).x9 = (1 succ 1).(<:<*n proj i*>:>.x9)
    by A8,FUNCT_1:13
      .= (1 succ 1).<*(n proj i).x9*> by A7,FINSEQ_3:141
      .= (1 succ 1).<*x9.i*> by Th37
      .= (ii.1)+1 by Def7
      .= xi+1;
    hence (n succ i).x = ((1 succ 1)*<:<*n proj i*>:>).x by A10;
  end;
  then
A11: n succ i = (1 succ 1)*<:<*n proj i*>:>;
A12: rng <*n proj i*> c= P
  proof
    let x be object;
    assume x in rng <*n proj i*>;
    then x in {n proj i} by FINSEQ_1:39;
    then x = n proj i by TARSKI:def 1;
    hence thesis by A3,A4,Def14;
  end;
  P is composition_closed by Def14;
  hence thesis by A11,A1,A2,A12,A5;
end;
