reserve x,y for object,
        i,j,k,m,n for Nat;

theorem Th4:
  for f be odd-valued FinSequence
    for o be DoubleReorganization of dom f st
     for n holds 2*n-1 = f.o_(n,1) & ... & 2*n-1 = f.o_(n,len (o.n))
  holds o is odd_organization of f
proof
  let f be odd-valued FinSequence;
  let o be DoubleReorganization of dom f such that A1:
     for n holds 2*n-1 = f.o_(n,1) & ... & 2*n-1 = f.o_(n,len (o.n));
A2:  for n ex x st
      x = f.o_(n,1) & ... & x = f.o_(n,len (o.n))
  proof
    let n;
    take x=2*n-1;
    thus thesis by A1;
  end;
  for n1,n2,i1,i2 be Nat st
    i1 in dom (o.n1) & i2 in dom (o.n2) &
      f.o_(n1,i1) = f.o_(n2,i2)
    holds n1 = n2
  proof
    let n1,n2,i1,i2 be Nat such that
    A3: i1 in dom (o.n1) & i2 in dom (o.n2) &
    f.o_(n1,i1) = f.o_(n2,i2);
    A4:2*n1-1 = f.o_(n1,1) & ... & 2*n1-1 = f.o_(n1,len (o.n1)) by A1;
    1<= i1 & i1 <= len (o.n1) by A3,FINSEQ_3:25;
    then A5: f.o_(n1,i1) = 2*n1-1 by A4;
    A6:2*n2-1 = f.o_(n2,1) & ... & 2*n2-1 = f.o_(n2,len (o.n2)) by A1;
    1<= i2 & i2 <= len (o.n2) by A3,FINSEQ_3:25;
    then 2*n2-1 = 2*n1-1 by A6,A5,A3;
    hence thesis;
  end;
  then o is valued_reorganization of f by A2,FLEXARY1:def 9;
  hence thesis by A1,Def4;
end;
