reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem Th26:
  for O being Ordinal,
      L being non empty ZeroStr,
      s being Series of O,L,
   perm being Permutation of O holds
      rng (s permuted_by perm) = rng s
proof
  let O be Ordinal,L be non empty ZeroStr,
    s be Series of O,L,perm be Permutation of O;
  set P=s permuted_by perm;
A1:dom P = Bags O & dom s= Bags O by FUNCT_2:def 1;
  thus rng P c= rng s
  proof
    let y be object;
    assume y in rng P;
    then consider x be object such that
A2: x in dom P & P.x=y by FUNCT_1:def 3;
    reconsider x as Element of Bags O by A2;
A3:   x*perm in dom s by A1,PRE_POLY:def 12;
    s.(x*perm) = P.x by Def4;
    hence thesis by A3,FUNCT_1:def 3,A2;
  end;
  let y be object;
  assume y in rng s;
  then consider x be object such that
A4: x in dom s & s.x=y by FUNCT_1:def 3;
  reconsider x as Element of Bags O by A4;
A5: dom x = O by PARTFUN1:def 2;
  dom perm = O by FUNCT_2:52;
  then perm"*perm = id O by FUNCT_1:39;
  then (x*perm")*perm = x*(id O) by RELAT_1:36
  .= x by A5,RELAT_1:51;
  then
A6:P.(x*perm") = s.x by Def4;
  x*perm" in dom P by A1,PRE_POLY:def 12;
  hence thesis by A4,A6,FUNCT_1:def 3;
end;
