
theorem Th21:
for L being well-unital add-associative right_zeroed right_complementable
            Abelian associative distributive non degenerated doubleLoopStr
for p being non zero Polynomial of L
holds deg even_part(p) <> deg odd_part(p)
proof
let L be add-associative right_zeroed right_complementable associative
         Abelian well-unital distributive non degenerated doubleLoopStr;
let p be non zero Polynomial of L;
set e = even_part(p), o = odd_part(p);
per cases;
suppose A1: o = 0_.(L) or e = 0_.(L);
  then A2: deg o = - 1 or deg e = - 1 by HURWITZ:20;
  A3: 0_.(L) + 0_.(L) = 0_.(L) by POLYNOM3:28;
  now per cases by A1;
  case o = 0_.(L);
    then e <> 0_.(L) by A3,Th9;
    hence thesis by A2,HURWITZ:20;
    end;
  case e = 0_.(L);
    then o <> 0_.(L) by A3,Th9;
    hence thesis by A2,HURWITZ:20;
    end;
  end;
  hence thesis;
  end;
suppose o <> 0_.(L) & e <> 0_.(L);
  then reconsider e,o as non zero Polynomial of L by UPROOTS:def 5;
  reconsider de = degree e as Element of NAT by ORDINAL1:def 12;
  reconsider deo = degree o as Element of NAT by ORDINAL1:def 12;
  now assume A4: deg e = deg o;
    degree e + 1 = len e;
    then A5: e.de <> 0.L by ALGSEQ_1:10;
    degree o + 1 = len o;
    then A6: o.deo <> 0.L by ALGSEQ_1:10;
    now per cases;
    case degree e is even;
      hence contradiction by A6,A4,Def2;
      end;
    case degree e is odd;
      hence contradiction by A5,Def1;
      end;
    end;
    hence contradiction;
    end;
  hence thesis;
  end;
end;
