
theorem lemlowp1a2:
for R being non degenerated Ring,
    p,q being non zero Polynomial of R
st p + q <> 0_.(R) &
   min* {i where i is Nat : p.i <> 0.R} = min* {i where i is Nat : q.i <> 0.R}
holds min* {i where i is Nat : (p+q).i <> 0.R} >=
                                          min* {i where i is Nat : p.i <> 0.R}
proof
let R be non degenerated Ring, p,q be non zero Polynomial of R;
assume XX: p + q <> 0_.(R) & min* {i where i is Nat : p.i <> 0.R} =
                             min* {i where i is Nat : q.i <> 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 non empty Subset of NAT by lemlp0,XX,TARSKI:def 3;
min* cpq in cpq by NAT_1:def 1;
then consider u being Nat such that H1: u = min* cpq & (p+q).u <> 0.R;
  now let j be Nat;
      assume D0: j < min* cp;
      D1a: now assume p.j <> 0.R;
           then j in cp;
           hence contradiction by D0,NAT_1:def 1;
           end;
      now assume q.j <> 0.R;
        then j in cq;
        hence contradiction by XX,D0,NAT_1:def 1;
        end;
      hence (p+q).j = p.j + 0.R by NORMSP_1:def 2 .= 0.R by D1a;
      end;
  hence thesis by H1;
end;
