let D be non empty set ; :: thesis: for n being Element of NAT st D is countable holds
n -tuples_on D is countable
let n be Element of NAT ; :: thesis: ( D is countable implies n -tuples_on D is countable )
defpred S₁[ Nat] means $1 -tuples_on D is countable ;
assume card D c= omega ; :: according to CARD_3:def 14 :: thesis: n -tuples_on D is countable
then card (1 -tuples_on D) c= omega by Th56;
then A1: 1 -tuples_on D is countable by CARD_3:def 14;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then [:(k -tuples_on D),(1 -tuples_on D):] is countable by A1, Th55;
then A3: card [:(k -tuples_on D),(1 -tuples_on D):] c= omega by CARD_3:def 14;
k in NAT by ORDINAL1:def 12;
then card [:(k -tuples_on D),(1 -tuples_on D):] = card ((k + 1) -tuples_on D) by Th57;
hence S₁[k + 1] by A3, CARD_3:def 14; :: thesis: verum

end;

{(<*> D)} is countable ;
then A4: S₁[ 0 ] by FINSEQ_2:94;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A2);
hence n -tuples_on D is countable ; :: thesis: verum