 reserve o,o1,o2 for object;
 reserve n for Ordinal;
 reserve R,L for non degenerated comRing;
 reserve b for bag of 1;

theorem Th50:
    for R holds (BSPoly(R)).(1.Polynom-Ring(1,R)) = 1.(Polynom-Ring R)
    proof
      let R;
      set FS = Formal-Series R;
      reconsider PR = Polynom-Ring R as Subring of FS by Th35;
A1:   1.PR = 1_.(R) by POLYNOM3:def 10 .= 1.FS by POLYALG1:def 2;
      1.Formal-Series(1,R) = 1_(1,R) by Def3
      .= 1.Polynom-Ring(1,R) by POLYNOM1:def 11; then
      (BSPoly(R)).(1.Polynom-Ring(1,R))
      = (BSFSeri(R)).(1.Formal-Series(1,R)) by FUNCT_1:49
      .= 1.Polynom-Ring R by A1,Th30;
      hence thesis;
   end;
