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
  for f be finite Function
    for o be valued_reorganization of f holds
      rng (f*.o.n) = {} or (rng (f*.o.n) = {f.o_(n,1)} & 1 in dom (o.n))
  proof
    let f be finite Function;
    let o be valued_reorganization of f;
    assume rng (f*.o.n) <> {};
    then consider y such that
    A1:y in rng (f*.o.n) by XBOOLE_0:def 1;
    consider x such that
    A2:x in dom (f*.o.n) & (f*.o.n).x = y by A1,FUNCT_1:def 3;
    reconsider x as Nat by A2;
    A3:dom (f*.o) = dom o by FOMODEL2:def 6;
    n in dom (f*.o)
    proof
      assume not n in dom (f*.o);
      then (f*.o).n={} by FUNCT_1:def 2;
      hence thesis by A1;
    end;
    then (f*.o.n) = f*(o.n) by A3, FOMODEL2:def 6;
    then A4: (f*(o.n)).x = f.(o.n.x) & x in dom (o.n) by A2,FUNCT_1:11,12;
    consider w be object such that
    A5:w = f.o_(n,1) & ... & w = f.o_(n,len (o.n)) by Def9;
    1<= x & x <= len (o.n) by A4,FINSEQ_3:25;
    then A6:w = f.o_(n,x) & 1<= len (o.n) by XXREAL_0:2,A5;
    rng (f*.o.n) c= {f.o_(n,1)}
    proof
      let z be object;
      assume A7:z in rng (f*.o.n);
      then consider x such that
      A8:x in dom (f*.o.n) & (f*.o.n).x = z by FUNCT_1:def 3;
      reconsider x as Nat by A8;
      A9:dom (f*.o) = dom o by FOMODEL2:def 6;
      n in dom (f*.o)
      proof
        assume not n in dom (f*.o);
        then (f*.o).n={} by FUNCT_1:def 2;
        hence thesis by A7;
      end;
      then A10: (f*.o.n) = f*(o.n) by A9, FOMODEL2:def 6;
      then A11: (f*(o.n)).x = f.(o.n.x) & x in dom (o.n) by A8,FUNCT_1:11,12;
      then 1<= x & x <= len (o.n) by FINSEQ_3:25;
      then w = f.o_(n,x) & 1<= len (o.n) by XXREAL_0:2,A5;
      then z = f.o_(n,1) by A5,A8,A11,A10;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis by A6,FINSEQ_3:25,ZFMISC_1:33;
  end;
