
theorem
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed non trivial addLoopStr, p
  being Polynomial of n,L holds HM(p,T) + Red(p,T) = p
proof
  let n be Ordinal, O being connected TermOrder of n, L be add-associative
right_complementable right_zeroed non trivial addLoopStr, p be Polynomial of
  n,L;
A1: now
    let x be object;
    assume x in Bags n;
    then reconsider x9 = x as Element of Bags n;
    now
      per cases;
      case
A2:     x = HT(p,O);
        hence (HM(p,O) + Red(p,O)).x = HM(p,O).HT(p,O) + Red(p,O).HT(p,O) by
POLYNOM1:15
          .= HM(p,O).HT(p,O) + 0.L by Lm18
          .= HM(p,O).HT(p,O) by RLVECT_1:4
          .= p.x by A2,Lm8;
      end;
      case
A3:     x <> HT(p,O);
        (HM(p,O) + Red(p,O)).x9 = HM(p,O).x9 + Red(p,O).x9 by POLYNOM1:15
          .= HM(p,O).x9 + p.x9 by A3,Lm19
          .= 0.L + p.x9 by A3,Th19
          .= p.x9 by RLVECT_1:4;
        hence p.x = (HM(p,O) + Red(p,O)).x;
      end;
    end;
    hence p.x = (HM(p,O) + Red(p,O)).x;
  end;
  dom p = Bags n & dom(HM(p,O) + Red(p,O)) = Bags n by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
