
theorem Th8:
  for L be add-associative right_zeroed right_complementable non
  empty addLoopStr for p be Polynomial of L holds len (-p) = len p
proof
  let L be add-associative right_zeroed right_complementable non empty
  addLoopStr;
  let p be Polynomial of L;
A1: now
    let n be Nat;
    assume
A2: n is_at_least_length_of -p;
    n is_at_least_length_of p
    proof
      let i be Nat;
      assume i >= n;
      then i in NAT & (-p).i = 0.L by A2,ORDINAL1:def 12;
      then -p.i = 0.L by BHSP_1:44;
      hence p.i = 0.L by VECTSP_2:3;
    end;
    hence len p <= n by ALGSEQ_1:def 3;
  end;
  len p is_at_least_length_of -p
  proof
    let i be Nat;
    assume
A3: i >= len p;
    thus (-p).i = -(p.i) by BHSP_1:44
      .= -0.L by A3,ALGSEQ_1:8
      .= 0.L by RLVECT_1:12;
  end;
  hence thesis by A1,ALGSEQ_1:def 3;
end;
