 reserve a for non empty set;
 reserve b, x, o for object;
reserve R for right_zeroed add-associative right_complementable Abelian
  well-unital distributive associative non trivial non trivial doubleLoopStr;
reserve R for non degenerated comRing;

theorem Th22:
   for p0 be Element of Polynom-Ring(0,R) holds
   not p0 is Polynomial of Polynom-Ring(0,R)
   proof
     let p0 be Element of Polynom-Ring(0,R);
reconsider P0 = p0 as Series of 0,R by POLYNOM1:def 11;
     not p0 in [#]Polynom-Ring(Polynom-Ring(0,R))
     proof
       assume
       p0 in [#]Polynom-Ring(Polynom-Ring(0,R)); then
reconsider P1 = p0 as sequence of Polynom-Ring(0,R) by POLYNOM3:def 10;
A2:    dom P1 = NAT by FUNCT_2:def 1;
       1 --> 0 in Bags 1 by PRE_POLY:def 12; then
       dom P0 <> dom P1 by A2;
       hence contradiction;
     end;
     hence thesis by POLYNOM3:def 10;
   end;
