
theorem
for L being add-associative right_zeroed right_complementable
      non empty addLoopStr,
    p being Polynomial of L
holds p - even_part p = odd_part p
proof
let L be add-associative right_zeroed right_complementable
     non empty addLoopStr,
    p be Polynomial of L;
set e = even_part p, o = odd_part p;
A1: dom(p-e) = NAT by FUNCT_2:def 1 .= dom o by FUNCT_2:def 1;
now let x be object;
  assume x in dom o;
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  p = o + e by Th10;
  then p.i = o.i + e.i by NORMSP_1:def 2;
  then p.i - e.i = o.i + (e.i + - e.i) by RLVECT_1:def 3
                .= o.i + 0.L by RLVECT_1:5
                .= o.i by RLVECT_1:def 4;
  hence (p-e).x = o.x by POLYNOM3:27;
  end;
hence thesis by A1,FUNCT_1:2;
end;
