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

theorem Th22:
  for O being Ordinal,
      L being non empty ZeroStr,
      perm being Permutation of O,
      s being Series of O,L, b being bag of O holds
       b*perm" in Support (s permuted_by perm) iff b in Support s
proof
  let O be Ordinal,
      L be non empty ZeroStr,
      perm be Permutation of O,
      s be Series of O,L, b be bag of O;
  set P = s permuted_by perm;
A1: dom P= Bags O = dom s by FUNCT_2:def 1;
A2: dom b = O by PARTFUN1:def 2;
  dom perm = O by FUNCT_2:52;
  then perm"*perm = id O by FUNCT_1:39;
  then (b*perm")*perm = b*(id O) by RELAT_1:36 .= b by A2,RELAT_1:51;
  then
A3:P.(b*perm") = s.b by Def4;
  thus b*perm" in Support P implies b in Support s
  proof
    assume b*perm" in Support P;
    then
A4: P.(b*perm") <>0.L by POLYNOM1:def 3;
    b in Bags O by PRE_POLY:def 12;
    hence thesis by A4,A1,A3,POLYNOM1:def 3;
  end;
  assume b in Support s;
  then
A5: s.b<>0.L by POLYNOM1:def 3;
  b*perm" in Bags O by PRE_POLY:def 12;
  hence thesis by A5,A1,A3,POLYNOM1:def 3;
end;
