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

theorem Th48:
    for f,g be Element of Polynom-Ring(1,R) holds
      (BSPoly(R)).(f+g) = (BSPoly(R)).f + (BSPoly(R)).g
    proof
      let f,g be Element of Polynom-Ring(1,R);
      set FS1 = Formal-Series(1,R),
          FS = Formal-Series R;
      reconsider PR = Polynom-Ring R as Subring of FS by Th35;
reconsider f0 = f, g0 = g as Element of FS1 by Lm7,TARSKI:def 3;
A1:   (BSPoly(R)).f = (BSFSeri(R)).f by FUNCT_1:49;
A2:   (BSPoly(R)).g = (BSFSeri(R)).g by FUNCT_1:49;
A3:   (BSPoly(R)).(f+g) = (BSFSeri(R)).(f+g) by FUNCT_1:49;
reconsider s = (BSFSeri(R)).f0, t = (BSFSeri(R)).g0 as Element of FS;
reconsider p = (BSPoly(R)).f, q = (BSPoly(R)).g as Element of PR;
      (BSPoly(R)).f + (BSPoly(R)).g = p + q .= s + t by A1,A2,Lm3
      .= (BSFSeri(R)).(f0+g0) by Th27 .= (BSPoly(R)).(f+g) by Th46,A3;
      hence thesis;
    end;
