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
  i-tuples_on D = j-tuples_on A implies i = j
proof
  set f = i |-> the Element of D;
  f is Tuple of i,D by Th61;
  then
A1:  f in i-tuples_on D by Lm6;
  assume i-tuples_on D = j-tuples_on A;
   then f in j-tuples_on A by A1;
   then f is j-element by Lm7;
   then len f = j;
  hence thesis by CARD_1:def 7;
end;
