
theorem T11a:
for R being Ring,
    a being Element of R holds deg(a|R) = 0 iff a <> 0.R
proof
let L be Ring, a be Element of L;
now assume Y: a <> 0.L;
   now assume deg(a|L) < 0;
     then a|L = 0_.(L) by T8b .= (0.L)|L by T6;
     hence contradiction by Y,T9;
     end;
   hence deg(a|L) = 0 by RATFUNC1:def 2;
   end;
hence thesis by T8b,T6;
end;
