
theorem T11:
for R being Ring
for p being Element of the carrier of Polynom-Ring R
holds p is constant iff ex a being Element of R st p = a|R
proof
let L be Ring, p be Element of the carrier of Polynom-Ring L;
reconsider g = p as Polynomial of L;
now assume p is constant;
   then AS: (len p) - 1 + 1 <= 0+1 by XREAL_1:6;
   per cases by AS,NAT_1:25;
   suppose len p = 0;
     then p = 0_.(L) by POLYNOM4:5 .= (0.L)|L by T6;
     hence ex a being Element of L st p = a|L;
     end;
   suppose AS: len p = 1;
     set q = (p.0)|L;
     now let x be object;
       assume x in NAT;
       then reconsider i = x as Element of NAT;
       per cases;
       suppose i = 0;
         hence q.x = p.x by Th28;
         end;
       suppose B: i <> 0;
         then i + 1 > 0 + 1 by XREAL_1:6;
         then i >= len p by AS,NAT_1:13;
         hence p.x = 0.L by ALGSEQ_1:8 .= q.x by B,Th28;
         end;
       end;
    hence ex a being Element of L st p = a|L by FUNCT_2:12;
     end;
   end;
hence thesis by RATFUNC1:def 2;
end;
