
theorem Th10:
for L being left_zeroed right_zeroed non empty addLoopStr,
    p being Polynomial of L
holds odd_part p + even_part p = p
proof
let L be left_zeroed right_zeroed non empty addLoopStr,
    p be Polynomial of L;
set e = even_part p, o = odd_part p;
A1: dom p = NAT by FUNCT_2:def 1 .= dom(o+e) by FUNCT_2:def 1;
now let x be object;
  assume x in dom p;
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case A2: i is even;
    hence o/.i + e/.i = 0.L + e.i by Def2
                     .= e.i by ALGSTR_1:def 2 .= p/.i by Def1,A2;
    end;
  case A3: i is odd;
    hence o/.i + e/.i = o.i + 0.L by Def1
                     .= o.i by RLVECT_1:def 4 .= p/.i by Def2,A3;
    end;
  end;
  hence p.x = (o+e).x by NORMSP_1:def 2;
  end;
hence thesis by A1,FUNCT_1:2;
end;
