
theorem Th25:
  for L be add-associative right_zeroed right_complementable
  well-unital commutative associative distributive almost_left_invertible non
empty doubleLoopStr for p be Polynomial of L for v be Element of L st v <> 0.L
  holds len (v*p) = len p
proof
  let L be add-associative right_zeroed right_complementable well-unital
  commutative associative distributive almost_left_invertible non empty
  doubleLoopStr;
  let p be Polynomial of L;
  let v be Element of L;
  assume
A1: v <> 0.L;
A2: now
    let n be Nat;
    assume
A3: n is_at_least_length_of v*p;
    n is_at_least_length_of p
    proof
      let i be Nat;
      reconsider i1=i as Element of NAT by ORDINAL1:def 12;
      assume i >= n;
      then (v*p).i = 0.L by A3;
      then v*p.i1 = 0.L by Def4;
      hence thesis by A1,VECTSP_1:12;
    end;
    hence len p <= n by ALGSEQ_1:def 3;
  end;
  len p is_at_least_length_of (v*p)
  proof
    let i be Nat;
    assume
A4: i >= len p;
    reconsider ii=i as Element of NAT by ORDINAL1:def 12;
    thus (v*p).i = v*p.ii by Def4
      .= v*0.L by A4,ALGSEQ_1:8
      .= 0.L;
  end;
  hence thesis by A2,ALGSEQ_1:def 3;
end;
