
theorem lemlp0:
for R being Ring,
    p being Polynomial of R
holds p = 0_.(R) iff {i where i is Nat : p.i <> 0.R} = {}
proof
let R be Ring, p be Polynomial of R;
hereby assume AS: p = 0_.(R);
   now assume {i where i is Nat : p.i <> 0.R} is non empty;
     then consider x being object such that
     H1: x in {i where i is Nat : p.i <> 0.R};
     consider i being Nat such that H2: x = i & p.i <> 0.R by H1;
     thus contradiction by AS,H2,FUNCOP_1:7,ORDINAL1:def 12;
     end;
   hence {i where i is Nat : p.i <> 0.R} = {};
   end;
assume AS: {i where i is Nat : p.i <> 0.R} = {};
  assume p <> 0_.(R);
    then len p <> 0 by POLYNOM4:5;
    then len p + 1 > 0 + 1 by XREAL_1:6;
    then len p >= 1 by NAT_1:13;
    then reconsider k = len p - 1 as Element of NAT by INT_1:3;
    reconsider k as Nat;
    len p = k + 1;
    then p.k <> 0.R by ALGSEQ_1:10;
    then k in {i where i is Nat : p.i <> 0.R};
    hence contradiction by AS;
end;
