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*;
reserve m,n for Nat,
  s,w for FinSequence of NAT;

theorem
 for z being set holds z is Tuple of i,D iff z in i-tuples_on D
 proof let z be set;
  thus z is Tuple of i,D implies z in i-tuples_on D by Lm6;
  assume z in i-tuples_on D;
   then ex s being Element of D* st z = s & len s = i;
  hence z is Tuple of i,D by CARD_1:def 7;
 end;
