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

theorem Th1:
   for f be sequence of R holds
   Support f = {} iff f = 0_.R
   proof
     let f be sequence of R;
     f = 0_.R implies Support f = {}
     proof
       assume
A1:    f = 0_.R;
       Support f = {}
       proof
         assume Support f <> {}; then
         consider o such that
A2:      o in Support f by XBOOLE_0:def 1;
         reconsider x1 = o as Element of NAT by A2;
         f.x1 <> 0.R by A2,POLYNOM1:def 4;
         hence contradiction by A1;
       end;
       hence thesis;
     end;
     hence thesis by POLYNOM1:def 4;
   end;
