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 Th9:
  1 <= i & i <= n+1 implies
  for p being Element of (n+1)-tuples_on D holds Del(p,i) in n-tuples_on D
proof
  set X = D;
  assume that
A1: 1 <= i and
A2: i <= n+1;
  let p be Element of (n+1)-tuples_on X;
A3: len p = n+1 by CARD_1:def 7;
  then i in dom p by A1,A2,FINSEQ_3:25;
  then
A4: len Del(p,i) = n by A3,FINSEQ_3:109;
  Del(p,i) is FinSequence of X by FINSEQ_3:105;
  then Del(p,i) is Element of n-tuples_on X by A4,FINSEQ_2:92;
  hence thesis;
end;
