reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th34:
  {} is DoubleReorganization of {} & <* {} *> is DoubleReorganization of {}
proof
  {} = <*> ({}*);
  then reconsider D={} as double-one-to-one FinSequence of {}*;
  rngs D = {} --> D by FUNCT_6:23;
  then Union rngs D={} by FUNCT_6:26;
  then Values D = {} by MATRIX_0:def 9;
  hence {} is DoubleReorganization of {} by Def7;
  rng {} = {};
  then reconsider F={} as FinSequence of {} by FINSEQ_1:def 4;
  {F} c= {}* & rng <*F*> = {F} by FINSEQ_1:38;
  then reconsider FF=<*F*> as double-one-to-one FinSequence of {}*
    by FINSEQ_1:def 4;
  A1: rngs FF = <*rng F*> by FINSEQ_3:132;
  rng <*rng F*> = {rng F} by FINSEQ_1:38;
  then union rng <*rng F*> =rng F by ZFMISC_1:25;
  then Union rngs FF = rng F by CARD_3:def 4,A1;
  then Values FF ={} by MATRIX_0:def 9;
  hence thesis by Def7;
end;
