reserve i,j,n,n1,n2,m,k,l,u for Nat,
        i1,i2,i3,i4,i5,i6 for Element of n,
        p,q for n-element XFinSequence of NAT,
        a,b,c,d,e,f for Integer;

theorem
  for n being Ordinal,L being add-associative right_zeroed right_complementable
                              non empty addLoopStr,
      p being Polynomial of n, L holds
  degree p = 0 iff Support p c= {EmptyBag n}
proof
  let n be Ordinal, L be add-associative right_zeroed right_complementable
                         non empty addLoopStr,
  p be Polynomial of n, L;
  thus degree p = 0 implies Support p c= {EmptyBag n}
    proof
      assume
A1:     degree p = 0;
      per cases;
      suppose
A2:       p= 0_(n,L);
        let y be object;
        assume
A3:       y in Support p;
        then p.y<>0.L by POLYNOM1:def 3;
        hence thesis by A3,A2,POLYNOM1:22;
      end;
      suppose
A4:       p <> 0_(n,L);
        let y be object;
        assume
A5:       y in Support p;
        then reconsider S=y as bag of n;
        degree S= 0 by A1,A4,Def3,A5;
        then S=EmptyBag n by UPROOTS:12;
        hence y in {EmptyBag n} by TARSKI:def 1;
      end;
  end;
  assume
A6: Support p c= {EmptyBag n};
  assume
A7: degree p <>0;
  then p <>0_(n,L) by Def3;
  then consider s be bag of n such that
A8:  s in Support p & degree p = degree s by Def3;
  s=EmptyBag n by A6,A8,TARSKI:def 1;
  hence thesis by A7,A8,UPROOTS:11;
end;
