reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;

theorem
  for F being Function of [:D,D9:],E for z being Tuple of i,D
    holds F[:](z,d9) is Element of i-tuples_on E
proof
  let F be Function of [:D,D9:],E;
  let z be Tuple of i,D;
  reconsider r = F[:](z,d9) as FinSequence of E by Th81;
  len z = i by CARD_1:def 7;
  then len r = i by Th82;
  hence thesis by Th90;
end;
