
theorem lemlowp2:
for R being non degenerated Ring,
    p being Polynomial of R holds
min* {i where i is Nat : (-p).i <> 0.R}
                                = min* {i where i is Nat : p.i <> 0.R}
proof
let R be non degenerated Ring, p be Polynomial of R;
per cases;
suppose p = 0_.(R);
  then -p = p by HURWITZ:9;
  hence thesis;
  end;
suppose p <> 0_.(R);
  then reconsider pp = p as non zero Polynomial of R by UPROOTS:def 5;
  reconsider cp = {i where i is Nat : pp.i <> 0.R} as
                             non empty Subset of NAT by lemlp1;
  min* cp in cp by NAT_1:def 1;
  then consider j being Nat such that A0: j = min* cp & p.j <> 0.R;
  now assume (-p).j = 0.R;
    then A1: 0.R = -(p.j) by BHSP_1:44;
    p.j = -(-p.j) .= 0.R by A1;
    hence contradiction by A0;
    end;
  then A: j in {i where i is Nat : (-p).i <> 0.R};
  now let o be object;
    assume o in {i where i is Nat : (-p).i <> 0.R};
    then consider i being Nat such that H1: o = i & (-p).i <> 0.R;
    thus o in NAT by H1,ORDINAL1:def 12;
    end;
  then reconsider cmp = {i where i is Nat : (-p).i <> 0.R} as
                           non empty Subset of NAT by A,TARSKI:def 3;
  now let k be Nat;
    assume k in {i where i is Nat : (-p).i <> 0.R};
    then consider i being Nat such that A3: k = i & (-p).i <> 0.R;
    now assume p.i = 0.R;
      then -(p.i) = 0.R;
      hence (-p).i = 0.R by BHSP_1:44;
      end;
    then i in cp by A3;
    hence j <= k by A3,A0,NAT_1:def 1;
    end;
  then j = min* cmp by A,NAT_1:def 1;
  hence thesis by A0;
  end;
end;
