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 z1 being Tuple of i,D
   for z2 being Tuple of i,D9 holds F.:(z1,z2) is Element of i
  -tuples_on E
proof
  let F be Function of [:D,D9:],E;
  let z1 be Tuple of i,D;
  let z2 be Tuple of i,D9;
  reconsider r = F.:(z1,z2) as FinSequence of E by Th68;
  len z1 = i & len z2 = i by CARD_1:def 7;
  then len r = i by Th70;
  hence thesis by Th90;
end;
