
theorem Th15:
for L being add-associative right_zeroed right_complementable
            Abelian non empty addLoopStr,
    p,q being Polynomial of L
holds even_part(p+q) = even_part(p) + even_part(q)
proof
let L be add-associative right_zeroed right_complementable
         Abelian non empty addLoopStr;
let p,q be Polynomial of L;
set epq = even_part(p+q),
    ep = even_part(p),
    eq = even_part(q);
A1: dom epq = NAT by FUNCT_2:def 1 .= dom(ep + eq) by FUNCT_2:def 1;
now let x be object;
  assume x in dom(epq);
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case A2: i is even;
     thus(ep + eq).i = ep.i + eq.i by NORMSP_1:def 2
                    .= p.i + eq.i by A2,Def1
                    .= p.i + q.i by A2,Def1
                    .= (p+q).i by NORMSP_1:def 2
                    .= (epq).i by A2,Def1;
    end;
  case A3: i is odd;
     thus (ep + eq).i = ep.i + eq.i by NORMSP_1:def 2
                    .= 0.L + eq.i by A3,Def1
                    .= 0.L + 0.L by A3,Def1
                    .= 0.L by RLVECT_1:def 4
                    .= (epq).i by A3,Def1;
    end;
  end;
  hence (ep + eq).x = (epq).x;
  end;
hence thesis by A1,FUNCT_1:2;
end;
