
theorem Th19:
for L being well-unital non empty doubleLoopStr
for p being Polynomial of L st deg p is even
holds odd_part Leading-Monomial(p) = 0_.(L)
proof
let L be well-unital non empty doubleLoopStr;
let p be Polynomial of L;
assume A1: deg p is even;
set LMp = Leading-Monomial(p);
set o = odd_part LMp;
A2: dom 0_.(L) = NAT by FUNCT_2:def 1 .= dom o by FUNCT_2:def 1;
now let x be object;
  assume x in dom 0_.(L);
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case len p = 0;
    then p = 0_.(L) by POLYNOM4:5;
    then LMp = 0_.(L) by POLYNOM4:13;
    hence o.x = (0_.(L)).x by Th7;
    end;
  case len p <> 0;
    then len p + 1 > 0 + 1 by XREAL_1:8;
    then len p >= 1 by NAT_1:13;
    then A3: len p -' 1 = deg p by XREAL_1:233;
    now per cases;
    case A4: i is odd;
      hence o.i = LMp.i by Def2
               .= 0.L by A4,A3,A1,POLYNOM4:def 1
               .= (0_.(L)).i by FUNCOP_1:7;
      end;
    case i is even;
      hence o.i = 0.L by Def2
               .= (0_.(L)).i by FUNCOP_1:7;
      end;
    end;
    hence o.x = (0_.(L)).x;
    end;
  end;
  hence o.x = (0_.(L)).x;
  end;
hence thesis by A2,FUNCT_1:2;
end;
