
theorem lemlowp1b:
for R being non degenerated Ring,
    p,q being non zero Polynomial of R
st p.(min* {i where i is Nat : p.i <> 0.R}) +
   q.(min* {i where i is Nat : q.i <> 0.R}) <> 0.R
holds min* {i where i is Nat : (p+q).i <> 0.R} =
                        min( min* {i where i is Nat : p.i <> 0.R},
                             min* {i where i is Nat : q.i <> 0.R})
proof
let R be non degenerated Ring, p,q be non zero Polynomial of R;
assume XX: p.(min* {i where i is Nat : p.i <> 0.R}) +
           q.(min* {i where i is Nat : q.i <> 0.R}) <> 0.R;
reconsider cp = {i where i is Nat : p.i <> 0.R},
           cq = {i where i is Nat : q.i <> 0.R} as non empty Subset of NAT
  by lemlp1;
now let o be object;
  assume o in {i where i is Nat : (p+q).i <> 0.R};
  then consider i being Nat such that H1: o = i & (p+q).i <> 0.R;
  thus o in NAT by H1,ORDINAL1:def 12;
  end;
then reconsider cpq = {i where i is Nat : (p+q).i <> 0.R}
     as Subset of NAT by TARSKI:def 3;
per cases by XXREAL_0:1;
suppose A: min* cp > min* cq;
  then min* cpq = min* cq by lemlowp1a1;
  hence thesis by A,XXREAL_0:def 9;
  end;
suppose A: min* cp < min* cq;
  then min* cpq = min* cp by lemlowp1a1;
  hence thesis by A,XXREAL_0:def 9;
  end;
suppose A: min* cp = min* cq;
  then (p+q).(min* cp) <> 0.R by XX,NORMSP_1:def 2;
  then D1: min* cp in cpq;
  then p + q <> 0_.(R) by lemlp0;
  then D4: min* cpq >= min* cp by A,lemlowp1a2;
  not(min* cpq > min* cp) by D1,NAT_1:def 1;
  hence thesis by A,D4,XXREAL_0:1;
  end;
end;
