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

theorem Th21:
    Polynom-Ring(n,R) is Subring of Formal-Series(n,R)
    proof
      set PR = Polynom-Ring(n,R),
          FS = Formal-Series(n,R),
      X = [: the carrier of PR, the carrier of PR:],
      Y = [: the carrier of FS, the carrier of FS:];
A1:   o in the carrier of PR implies o in the carrier of FS
      proof
        assume o in the carrier of PR; then
        o is Polynomial of n,R by POLYNOM1:def 11;
        hence thesis by Def3;
      end;
A2:   the carrier of PR c= the carrier of FS by A1;
A3:   the addF of PR = (the addF of FS)||(the carrier of PR)
      proof
A4:     dom (the addF of PR) = X by FUNCT_2:def 1;
        dom (the addF of FS) = Y by FUNCT_2:def 1; then
A5:     dom (the addF of FS) /\ X = X by XBOOLE_1:28, A2,ZFMISC_1:96;
        for o st o in dom (the addF of PR) holds
          (the addF of PR).o = (the addF of FS).o
        proof
          let o;
          assume o in dom the addF of PR; then
          consider o1,o2 such that
A6:       o1 in the carrier of PR & o2 in the carrier of PR &
          o = [o1,o2] by ZFMISC_1:def 2;
reconsider x = o1, y = o2 as Element of the carrier of PR by A6;
          reconsider s1 = x, s2 = y as finite-Support Series of n,R
          by POLYNOM1:def 11;
reconsider x1 = x, y1 = y as Element of the carrier of FS by A1;
          (the addF of PR).[x,y] = x + y
          .= s1 + s2 by POLYNOM1:def 11 .= x1 + y1 by Def3
          .= (the addF of FS).[x1,y1];
          hence thesis by A6;
        end;
        hence thesis by A4,A5,FUNCT_1:46;
      end;
A7:   the multF of PR = (the multF of FS)||(the carrier of PR)
      proof
A8:    dom (the multF of FS) = Y by FUNCT_2:def 1;
A9:    dom (the multF of PR) = X by FUNCT_2:def 1;
A10:    dom (the multF of FS) /\ X = X by A8,XBOOLE_1:28, A2,ZFMISC_1:96;
        for o st o in dom the multF of PR holds
        (the multF of PR).o = (the multF of FS).o
        proof
          let o;
          assume o in dom the multF of PR; then
          consider o1,o2 such that
A11:      o1 in the carrier of PR & o2 in the carrier of PR &
          o = [o1,o2] by ZFMISC_1:def 2;
reconsider x = o1, y = o2 as Element of the carrier of PR by A11;
          reconsider s1 = x, s2 = y as finite-Support Series of n,R
          by POLYNOM1:def 11;
reconsider x1 = x, y1 = y as Element of the carrier of FS by A1;
          (the multF of PR).[x,y] = x * y .= s1 *' s2 by POLYNOM1:def 11
          .= x1 * y1 by Def3.= (the multF of FS).[x1,y1];
          hence thesis by A11;
        end;
        hence thesis by A9,A10,FUNCT_1:46;
      end;
A12:  0.PR = 0_(n,R) by POLYNOM1:def 11 .= 0.FS by Def3;
      1.PR = 1_(n,R) by POLYNOM1:def 11 .= 1.FS by Def3;
      hence thesis by A1,TARSKI:def 3, A3,A7,A12,C0SP1:def 3;
    end;
