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;

theorem Th35:
  dom (n proj i) = n-tuples_on NAT &
  (1 <= i & i <= n implies rng (n proj i) = NAT)
proof
  thus
A1: dom (n proj i)= product(n|->NAT) by CARD_3:def 16
    .= n-tuples_on NAT by FINSEQ_3:131;
  assume that
A2: 1 <= i and
A3: i <= n;
  now
    let x be object;
    thus x in rng (n proj i) implies x in NAT
       by  ORDINAL1:def 12;
    assume x in NAT;
    then reconsider x9 = x as Element of NAT;
    reconsider d = n |-> x9 as FinSequence of NAT;
    i in Seg n by A2,A3,FINSEQ_1:1;
    then
A4: d.i = x9 by FUNCOP_1:7;
    (n proj i).d = d.i by A1,CARD_3:def 16;
    hence x in rng (n proj i) by A1,A4,FUNCT_1:def 3;
  end;
  hence thesis by TARSKI:2;
end;
