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
  D* = union the set of all i-tuples_on D where i is Nat
proof
  for a being object
   holds a in D* iff a in union the set of all i-tuples_on D where i is Nat
  proof let a be object;
    thus a in D* implies a in union the set of all
i-tuples_on D where i is Nat
    proof
      assume a in D*;
      then reconsider a as FinSequence of D by FINSEQ_1:def 11;
      a is Element of (len a)-tuples_on D & (len a)-tuples_on D in the set of
all i
      -tuples_on D where i is Nat  by Th90;
      hence thesis by TARSKI:def 4;
    end;
    assume a in union the set of all i-tuples_on D where i is Nat;
    then consider Z being set such that
A1: a in Z and
A2: Z in the set of all i-tuples_on D where i is Nat
    by TARSKI:def 4;
    consider i being Nat such that
A3: Z = i-tuples_on D by A2;
     ex s being Element of D* st s = a & len s = i by A1,A3;
    hence thesis;
  end;
  hence thesis by TARSKI:2;
end;
