
theorem lemir:
for F being Field
for p being irreducible Element of the carrier of Polynom-Ring F
for q being monic Element of the carrier of Polynom-Ring F
st q divides p holds q = 1_.(F) or q = NormPolynomial p
proof
let F be Field;
let p be irreducible Element of the carrier of Polynom-Ring F;
let q be monic Element of the carrier of Polynom-Ring F;
assume AS: q divides p; then
per cases by RING_4:41;
suppose deg q < 1; then
  deg q + 1 <= 1 by INT_1:7; then
  (deg q + 1) - 1 <= 1 - 1 by XREAL_1:9; then
  consider a being Element of F such that
  A: q = a|F by RING_4:def 4,RING_4:20;
  1.F = LC q by RATFUNC1:def 7 .= a by A,RING_5:6;
  hence thesis by A,RING_4:14;
  end;
suppose A: deg q >= deg p;
  deg q <= deg p by AS,RING_5:13; then
  B: deg q = deg p by A,XXREAL_0:1;
  consider r being Polynomial of F such that
  C: q *' r = p by AS,RING_4:1;
  r <> 0_.(F) by C; then
  deg p = deg p + deg r by C,B,HURWITZ:23; then
  r is constant Element of the carrier of Polynom-Ring F
     by RING_4:def 4,POLYNOM3:def 10; then
  consider a being Element of F such that D: r = a|F by RING_4:20;
  a = LC p
    proof
    reconsider lq = deg q as Element of NAT;
    consider s being FinSequence of the carrier of F such that
A2: len s = lq+1  and
A3: (q*'r).lq = Sum s and
A4: for k be Element of NAT st k in dom s
    holds s.k = q.(k-'1) * r.(lq+1-'k) by POLYNOM3:def 9;
A5: lq = len q - 1 by HURWITZ:def 2; then
A6: len q - 1 = len q -' 1 by XREAL_0:def 2;
    lq + 1 > 0; then
    len q >= 0 + 1 by A5,INT_1:7; then
    len q in Seg(len q) by FINSEQ_1:1; then
A7: len q in dom s by A2,A5,FINSEQ_1:def 3; then
A8: s/.(len q) = s.(len q) by PARTFUN1:def 6
              .= q.lq * r.((lq+1)-'(len q)) by A4,A5,A6,A7
              .= q.(deg q) * r.0 by A5,NAT_2:8
              .= LC q * r.0 by FIELD_6:2
              .= (1.F) * r.0 by RATFUNC1:def 7
              .= a by D,Th28;
    now let i be Element of NAT;
      assume
A9:   i in dom s & i <> len q; then
A10:  s.i = q.(i-'1) * r.((lq+1)-'i) by A4;
      i in Seg(len q) by A2,A5,A9,FINSEQ_1:def 3; then
      i <= len q by FINSEQ_1:1; then
      len q - i in NAT by INT_1:5; then
      (lq+1)-'i = len q - i by A5,XREAL_0:def 2; then
      (lq+1)-'i <> 0 by A9; then
      r.((lq+1)-'i) = 0.F by D,Th28;
      hence s/.i = 0.F by A9,A10,PARTFUN1:def 6;
      end;
    hence a = Sum s by A8,A7,POLYNOM2:3
           .= LC p by A3,C,B,FIELD_6:2;
    end; then
  E: r = (LC p) * 1_.(F) by D,RING_4:16;
  (LC p)" * p = q *' ((LC p)" * r) by C,RING_4:10
             .= q *' (((LC p)" * (LC p)) * 1_.(F)) by E,RING_4:11
             .= q *' ((1.F) * 1_.(F)) by VECTSP_1:def 10
             .= q;
  hence thesis by RING_4:23;
  end;
end;
