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 Th84:
  [+].<*i,j*> = i+j
proof
  reconsider i,j as Element of NAT by ORDINAL1:def 12;
  reconsider q = <*i*> as Element of 1-tuples_on NAT by FINSEQ_2:131;
  defpred p[Nat] means [+].<*i,$1*> = i+$1;
A1: now
    let j be Nat;
    reconsider jj=j as Element of NAT by ORDINAL1:def 12;
    reconsider r = <*i,jj,i+jj*> as Element of 3-tuples_on NAT by FINSEQ_2:104;
    assume p[j];
    then [+].<*i,jj+1*> = (3 succ 3).r by Th80
      .= (r.3)+1 by Def7
      .= i+(j+1);
    hence p[j+1];
  end;
  [+].<*i,0*> = (1 proj 1).q by Th78
    .= q.1 by Th37
    .= i+0;
  then
A2: p[0];
  for j being Nat holds p[j] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
