
theorem lemppolspl3a:
for F being Field
for p being non constant monic Polynomial of F
for q being non zero Polynomial of F
st p *' q is Ppoly of F holds p is Ppoly of F
proof
let F be Field;
let p be non constant monic Polynomial of F;
let q be non zero Polynomial of F;
assume AS: p *' q is Ppoly of F;
p <> 0_.(F) & q <> 0_.(F); then
A1: deg(p*'q) = deg p + deg q by HURWITZ:23;
deg p > 0 by RATFUNC1:def 2; then
A: deg(p*'q) >= 1 + 0 by A1,NAT_1:14;

defpred P[Nat] means
  for p being non constant monic Polynomial of F
  for q being non zero Polynomial of F
  st deg(p*'q) = $1 & p*'q is Ppoly of F holds p is Ppoly of F;

now let p be non constant monic Polynomial of F;
  let q be non zero Polynomial of F;
  assume B: deg(p*'q) = 1 & p*'q is Ppoly of F;
  p <> 0_.(F) & q <> 0_.(F); then
  deg(p*'q) = deg p + deg q by HURWITZ:23; then
  B1: deg p <= deg(p*'q) by NAT_1:12;
  deg p > 0 by RATFUNC1:def 2; then
  deg p >= deg(p*'q) by B,NAT_1:14; then
  consider x,z being Element of F such that
  A3: x <> 0.F & p = x * rpoly(1,z) by B,B1,XXREAL_0:1,HURWITZ:28;
  x * LC rpoly(1,z) = LC(x * rpoly(1,z)) by RING_5:5
                   .= 1.F by A3,RATFUNC1:def 7;
  then x * 1.F = 1.F by RATFUNC1:def 7;
  hence p is Ppoly of F by A3,RING_5:51;
  end; then
IA: P[1];
IS: now let k be Nat;
    assume BB: 1 <= k;
    assume IV: P[k];
    now let p be non constant monic Polynomial of F;
      let q be non zero Polynomial of F;
      assume B: deg(p*'q) = k+1 & p*'q is Ppoly of F; then
      LC p = 1.F & LC(p*'q) = 1.F by RATFUNC1:def 7; then
      B0: 1.F = 1.F * LC(q) by NIVEN:46; then
      B0a: q is monic by RATFUNC1:def 7;
      consider a being Element of F such that
      B1: a is_a_root_of (p*'q) by B,POLYNOM5:def 8;
      0.F = eval(p*'q,a) by B1,POLYNOM5:def 7
         .= eval(p,a) * eval(q,a) by POLYNOM4:24; then
      per cases by VECTSP_2:def 1;
      suppose eval(q,a) = 0.F; then
        consider s being Polynomial of F such that
        B2: q = rpoly(1,a) *' s by HURWITZ:33,POLYNOM5:def 7;
        B3: s is non zero by B2;
        B6: s <> 0_.(F) & p <> 0_.(F) by B2; then
        deg q = deg rpoly(1,a) + deg s by B2,HURWITZ:23
             .= 1 + deg s by HURWITZ:27; then
        B4: deg(p*'s) = deg p + (deg q - 1) by B6,HURWITZ:23
                     .= (deg p + deg q) - 1
                     .= deg(p*'q) - 1 by B6,HURWITZ:23
                     .= k by B;
        1.F = LC rpoly(1,a) * LC s by B0,B2,NIVEN:46
           .= 1.F * LC s by RING_5:9; then
        s is monic by RATFUNC1:def 7; then
        B5: p *' s is non constant monic by BB,B4,RATFUNC1:def 2;
        p *' q = rpoly(1,a) *' (p *' s) by B2,POLYNOM3:33; then
        p *' s is Ppoly of F by B,B5,lemppolspl3b;
        hence p is Ppoly of F by B3,B4,IV;
        end;
      suppose eval(p,a) = 0.F; then
        consider s being Polynomial of F such that
        B2: p = rpoly(1,a) *' s by POLYNOM5:def 7,HURWITZ:33;
        reconsider s as non zero Polynomial of F by B2;
        B6: s <> 0_.(F); then
        deg p = deg rpoly(1,a) + deg s by B2,HURWITZ:23
             .= 1 + deg s by HURWITZ:27; then
        B4: deg(q*'s) = deg q + (deg p - 1) by B6,HURWITZ:23
                     .= (deg p + deg q) - 1
                     .= deg(p*'q) - 1 by B6,HURWITZ:23
                     .= k by B;
        B5: p *' q = rpoly(1,a) *' (s *' q) by B2,POLYNOM3:33;
        B7b: 1.F = LC p by RATFUNC1:def 7
                .= LC rpoly(1,a) * LC s by B2,NIVEN:46
                .= 1.F * LC s by RING_5:9; then
        B7a: s is monic by RATFUNC1:def 7;
        reconsider s1 = s as Element of the carrier of Polynom-Ring F
                                                       by POLYNOM3:def 10;
        per cases;
        suppose K: deg s >= 1; then
          B8a: s is non constant by RATFUNC1:def 2;
          deg(s *' q) = deg s + deg q by B6,HURWITZ:23; then
          s *' q is non constant by K,RATFUNC1:def 2; then
          s *' q is Ppoly of F by B,B5,B7a,B0a,lemppolspl3b; then
          B9: s is Ppoly of F by B4,B7a,B8a,IV;
          rpoly(1,a) is Ppoly of F by RING_5:51;
          hence p is Ppoly of F by B9,B2,RING_5:52;
          end;
        suppose deg s < 1; then
          deg s = 0 by NAT_1:14; then
          consider b being Element of F such that
          B8: s1 = b|F by RING_4:def 4,RING_4:20;
          B9: s = b * 1_.(F) by B8,RING_4:16; then
          LC s = b * LC 1_.(F) by RING_5:5
              .= b * 1.F by RATFUNC1:def 7;
          hence p is Ppoly of F by B7b,B9,B2,RING_5:51;
          end;
        end;
      end;
    hence P[k+1];
    end;
  for i being Nat st 1 <= i holds P[i] from NAT_1:sch 8(IA,IS);
  hence thesis by A,AS;
end;
