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

theorem Th21:
  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 in Support (s permuted_by perm) iff b*perm 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:P.b = s.(b*perm) by Def4;
  thus b in Support P implies b*perm in Support s
  proof
    assume b in Support P;
    then
A3: P.b <>0.L by POLYNOM1:def 3;
    b*perm in Bags O by PRE_POLY:def 12;
    hence thesis by A3,A1,A2,POLYNOM1:def 3;
  end;
  assume b*perm in Support s;
  then
A4: s.(b*perm)<>0.L by POLYNOM1:def 3;
  b in Bags O by PRE_POLY:def 12;
  hence thesis by A4,A1,A2,POLYNOM1:def 3;
end;
