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;
