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

theorem Th42:
   for f be Series of 1,R holds BagN1.:(Support f) = Support(f*NBag1)
   proof
     let f be Series of 1,R;
     for o holds o in (BagN1).:(Support f) iff o in Support(f*NBag1)
     proof
       let o;
A1:    dom (BagN1) = Bags 1 by FUNCT_2:def 1;
A2:    dom (f*NBag1) = NAT by FUNCT_2:def 1;
A3:    o in BagN1.:(Support f) implies o in Support(f*NBag1)
       proof
         assume o in (BagN1).:(Support f); then
         consider x be object such that
A4:      x in dom (BagN1) & x in Support f & o = (BagN1).x by FUNCT_1:def 6;
A5:      o in NAT by A4,FUNCT_2:5;
         (f*NBag1).o = ((f*NBag1)*(BagN1)).x by A4,FUNCT_2:15
         .= f.x by Th38; then
         (f*NBag1).o <> 0.R by A4,POLYNOM1:def 3;
         hence thesis by A5,A2,POLYNOM1:def 3;
       end;
       o in Support(f*NBag1) implies o in BagN1.:(Support f)
       proof
         assume
A6:      o in Support(f*NBag1); then
        reconsider b = (NBag1).o as Element of Bags 1 by FUNCT_2:5;
A7:      dom f = Bags 1 by FUNCT_2:def 1;
A8:      BagN1.b = (id(NAT)).o by A6,FUNCT_2:15,Th10 .= o by A6,FUNCT_1:18;
         f.b = (f*(NBag1)).o by A6,FUNCT_2:15; then
         f.b <> 0.R by A6,POLYNOM1:def 3; then
         b in Support f by A7,POLYNOM1:def 3;
         hence thesis by A1,A8,FUNCT_1:def 6;
       end;
       hence thesis by A3;
     end;
     hence thesis by TARSKI:2;
   end;
