theorem
  for D being set holds 0-tuples_on D = { <*>D }
proof
  let D be set;
  now
    let z be object;
    thus z in 0-tuples_on D implies z = <*>D
    proof
      assume z in 0-tuples_on D;
      then ex s being Element of D* st z = s & len s = 0;
      hence thesis;
    end;
    <*>D is Element of D* & len <*>D = 0 by FINSEQ_1:def 11;
    hence z = <*>D implies z in 0-tuples_on D;
  end;
  hence thesis by TARSKI:def 1;
end;
