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 Th35:
  for D be finite set, F be one-to-one onto FinSequence of D holds
    <*F*> is DoubleReorganization of D
proof
  let D be finite set,
  F be one-to-one onto FinSequence of D;
  F is Element of D* by FINSEQ_1:def 11;
  then {F} c= D* & rng <*F*> = {F} by ZFMISC_1:31,FINSEQ_1:38;
  then reconsider FF=<*F*> as double-one-to-one FinSequence of D*
    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 = rng F by MATRIX_0:def 9;
  hence thesis by FUNCT_2:def 3,Def7;
end;
