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

theorem Th5:
  for f be odd-valued FinSequence,
      g be complex-valued FinSequence
    for o1,o2 be DoubleReorganization of dom g st
        o1 is odd_organization of f &
        o2 is odd_organization of f
       holds
         Sum (g*.o1).i =Sum (g*.o2).i
proof
  let f be odd-valued FinSequence,
      g be complex-valued FinSequence;
  A1:for o1,o2 be DoubleReorganization of dom g st
         o1 is odd_organization of f &
         o2 is odd_organization of f
       holds
     rng (f*.o1.n) c= rng (f*.o2.n)
  proof
    let o1,o2 be DoubleReorganization of dom g such that
    A2:o1 is odd_organization of f &
       o2 is odd_organization of f;
    reconsider O1=o1 as odd_organization of f by A2;
    let y be object;
    assume A3:y in rng (f*.o1.n);
    then A4: rng (f*.o1.n) = {f.o1_(n,1)} & 1 in dom (o1.n) by FLEXARY1:49,A2;
    A5:n in dom o1
    proof
      assume not n in dom o1;
      then o1.n={} by FUNCT_1:def 2;
      hence thesis by A3,FLEXARY1:49,A2;
    end;
    then o1.n.1 in Values o1 by FLEXARY1:1,A4;
    then o1.n.1 in dom f by FLEXARY1:def 7,A2;
    then o1.n.1 in Values o2 by FLEXARY1:def 7,A2;
    then consider j,w be object such that
    A6:j in dom o2 & w in dom (o2.j) & o2.j.w = o1.n.1 by FLEXARY1:1;
    reconsider j,w as Nat by A6;
    A7: (f*.o1).n = f*(o1.n) by A5,FOMODEL2:def 6;
    len ((f*.O1).n) = len (o1.n) by CARD_1:def 7;
    then A8: dom ((f*.o1).n) = dom (o1.n) by FINSEQ_3:29;
    A9:2*n-1 = f.o1_(n,1) & ... & 2*n-1 = f.o1_(n,len (o1.n)) by A2,Def4;
    A10: 1 <= len (o1.n) by A4,FINSEQ_3:25;
    A11:2*j-1 = f.o2_(j,1) & ... & 2*j-1 = f.o2_(j,len (o2.j)) by A2,Def4;
    1<= w & w <= len (o2.j) by A6,FINSEQ_3:25;
    then 2*j-1 = f.o2_(j,w) by A11;
    then A12: 2*j-1 = 2*n-1 by A6, A10,A9;
    then A13:y = f.o2_(n,w) by A4,A3,TARSKI:def 1,A6;
    A14:(f*.o2)_(n,w) = f.o2_(n,w) by FLEXARY1:42;
    A15:o2.n.w in dom f by A8,A4,A7,FUNCT_1:11,A6, A12;
    (f*.o2).n = f*(o2.n) by A12,A6,FOMODEL2:def 6;
    then w in dom ((f*.o2).n) by A15, A6, A12,FUNCT_1:11;
    hence thesis by FUNCT_1:def 3,A13,A14;
  end;
  let o1,o2 be DoubleReorganization of dom g such that
  A16:o1 is odd_organization of f &
      o2 is odd_organization of f;
  rng (f*.o1.i) = rng (f*.o2.i) by A1,A16;
  hence thesis by A16,FLEXARY1:51;
end;
