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 Th83:
  for g being 2-ary primitive-recursive Function holds
  (1,2)->(1,?,2) g in PrimRec
proof
A1: 3 proj 3 in PrimRec by Def14;
A2: 2-tuples_on PrimRec c= (PrimRec)* by FINSEQ_2:142;
  3 proj 1 in PrimRec by Def14;
  then <*3 proj 1, 3 proj 3*> in 2-tuples_on PrimRec by A1,FINSEQ_2:101;
  then reconsider F = <*3 proj 1, 3 proj 3*> as Element of (PrimRec)* by A2;
  F is with_the_same_arity
  proof
    let f,g be Function;
    assume that
A3: f in rng F and
A4: g in rng F;
A5: rng F = {3 proj 1, 3 proj 3} by FINSEQ_2:127;
    hence f is empty implies g is empty or dom g = {{}} by A3,TARSKI:def 2;
    assume that
    f is non empty and
    g is non empty;
    take 3, NAT;
    f = 3 proj 1 or f = 3 proj 3 by A3,A5,TARSKI:def 2;
    hence dom f c= 3-tuples_on NAT by Th35;
    g = 3 proj 1 or g = 3 proj 3 by A4,A5,TARSKI:def 2;
    hence thesis by Th35;
  end;
  then reconsider F as with_the_same_arity Element of (PrimRec)*;
  let g be 2-ary primitive-recursive Function;
  arity g = 2 by Def21;
  then
A6: arity g = len F by FINSEQ_1:44;
  g is Element of PrimRec by Def16;
  hence thesis by A6,Th71;
end;
