
theorem Th28:
  for L being add-associative right_zeroed right_complementable
  well-unital commutative associative distributive almost_left_invertible non
degenerated non empty doubleLoopStr for p being Polynomial of L holds deg(p)
  = 1 iff ex x,z being Element of L st x <> 0.L & p = x * rpoly(1,z)
proof
  let L be add-associative right_zeroed right_complementable well-unital
  commutative associative distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, p be Polynomial of L;
A1: now
    set x = p.1, z = (-p.0)*(p.1)";
    set f = x * rpoly(1,z);
    assume
A2: deg(p) = 1;
    then
A3: len p = 1 + 1;
    then
A4: x <> 0.L by ALGSEQ_1:10;
A5: now
      let k be Nat;
      assume k < len p;
      then k < 1+1 by A2;
      then
A6:   k <= 1 by NAT_1:13;
      per cases by A6,XXREAL_0:1;
      suppose
A7:     k = 1;
        hence f.k = x * rpoly(1,z).1 by POLYNOM5:def 4
          .= x * 1_L by Lm10
          .= p.k by A7;
      end;
      suppose
        k < 1;
        then
A8:     k = 0 by NAT_1:14;
        hence f.k = x * rpoly(1,z).0 by POLYNOM5:def 4
          .= x * (-(power(L).(z,1+0))) by Lm10
          .= x * (-(power(L).(z,0) * z)) by GROUP_1:def 7
          .= x * (-(1_L * z)) by GROUP_1:def 7
          .= x * (- z)
          .= p.1 * (-(-(p.0*(p.1)"))) by VECTSP_1:9
          .= p.1 * (p.0*(p.1)") by RLVECT_1:17
          .= (p.1 * (p.1)") * p.0 by GROUP_1:def 3
          .= 1_L * p.0 by A4,VECTSP_1:def 10
          .= p.k by A8;
      end;
    end;
    len f = deg(rpoly(1,z)) + 1 by A3,ALGSEQ_1:10,POLYNOM5:25
      .= 1+1 by Th27
      .= len p by A2;
    then p = f by A5,ALGSEQ_1:12;
    hence ex x,z being Element of L st x <> 0.L & p = x * rpoly(1,z) by A3,
ALGSEQ_1:10;
  end;
  now
    given x,z being Element of L such that
A9: x <> 0.L and
A10: p = x * rpoly(1,z);
    thus deg p = deg rpoly(1,z) by A9,A10,POLYNOM5:25
      .= 1 by Th27;
  end;
  hence thesis by A1;
end;
