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

theorem Th33:
  for R holds SBFSeri(R) is RingHomomorphism
  proof
    let R;
    set P = BSFSeri(R);
    set FS1 = Formal-Series(1,R);
    set FS = Formal-Series R;
A1: P is onto;
A2: P is additive multiplicative unity-preserving;
A3: for x,y being Element of FS holds
    P".(x+y) = P".x + P".y & P".(x*y) = P".x * P".y & P".(1_FS) = 1_FS1
    proof
A4:   P".(1_FS) = ((P qua Function)").(P.(1_FS1)) by GROUP_1:def 13
        .= 1_FS1 by FUNCT_2:26;
      let x,y be Element of FS;
      consider x9 being object such that
A5:   x9 in the carrier of FS1 and
A6:   P.(x9) = x by A1,FUNCT_2:11;
      reconsider x9 as Element of FS1 by A5;
A7:   x9 = ((P qua Function)").(P.(x9)) by FUNCT_2:26
        .= P".x by A6;
      consider y9 being object such that
A8:   y9 in the carrier of FS1 and
A9:   P.y9 = y by A1,FUNCT_2:11;
      reconsider y9 as Element of FS1 by A8;
A10:  y9 = ((P qua Function)").(P.(y9)) by FUNCT_2:26
      .= P".y by A9;
A11:  P".(x*y) = P".(P.(x9*y9)) by A2,A6,A9
      .= ((P qua Function)").(P.(x9*y9))
      .= P".x * P".y by A7,A10,FUNCT_2:26;
      P".(x+y) = P".(P.(x9+y9)) by A2,A6,A9
      .= ((P qua Function)").(P.(x9+y9))
      .= P".x + P".y by A7,A10,FUNCT_2:26;
      hence thesis by A11,A4;
    end;
    P" is additive multiplicative unity-preserving by A3;
    hence thesis by Th32;
  end;
