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

theorem Th2:
   for f be sequence of R st Support f is finite holds
   f is finite-Support sequence of R
   proof
     let f be sequence of R;
A1:  dom f = NAT by FUNCT_2:def 1;
     assume
A2:  Support f is finite;
     per cases;
       suppose Support f = {}; then
         f = 0_.R by Th1;
         hence thesis;
       end;
       suppose Support f <> {}; then
reconsider S = Support f as finite non empty Subset of NAT by A2;
reconsider m = max(S) as Element of NAT by ORDINAL1:def 12;
A3:    for s be Element of S holds m >= s by XXREAL_2:4;
       for i be Nat st i >= m+1 holds f.i = 0.R
       proof
         let i be Nat;
         assume
A4:      i >= m+1;
         assume
A5:      f.i <> 0.R;
         i in dom f by A1,ORDINAL1:def 12; then
         i in Support f by A5,POLYNOM1:def 4;
         hence contradiction by A4,A3,NAT_1:13;
       end;
       hence thesis by ALGSEQ_1:def 1;
       end;
     end;
