
theorem LX:
for L being add-associative right_zeroed right_complementable
            distributive non empty doubleLoopStr,
    p being Element of the carrier of Polynom-Ring L
holds p is non zero constant iff deg p = 0
proof
let L be add-associative right_zeroed right_complementable
         distributive non empty doubleLoopStr,
    p be Element of the carrier of Polynom-Ring L;
A: now assume AS: p is non zero constant;
   now assume deg p <> 0;
     then len p - 1 + 1 < 0 + 1 by AS,XREAL_1:6;
     then len p = 0 by NAT_1:14;
     then deg p = -1;
     then p = 0_.(L) by HURWITZ:20;
     hence contradiction by AS;
     end;
   hence deg p = 0;
   end;
now assume AS: deg p = 0;
  then p <> 0_.(L) by HURWITZ:20;
  hence p is non zero by UPROOTS:def 5;
  thus p is constant by AS;
  end;
hence thesis by A;
end;
