
theorem ThsepsplA:
for F being Field
for p being non constant Polynomial of F holds
deg p = card(Roots p) iff
(p splits_in F & for a being Element of F holds multiplicity(p,a) <= 1)
proof
let F be Field;
let p be non constant Polynomial of F;
A: now assume AS: deg p = card(Roots p);
   H: card(BRoots p) <= deg p by RING_5:42;
   deg p <= card(BRoots p) by AS,RING_5:65; then
   consider c being Element of F, q being Ppoly of F such that
   A: p = c * q by H,XXREAL_0:1,RING_5:66;
   p <> 0_.(F);
   then c is non zero  by A,POLYNOM5:26;
   hence p splits_in F &
         for a being Element of F holds multiplicity(p,a) <= 1
      by AS,A,H,ABC,FIELD_4:def 5;
   end;
now assume AS: p splits_in F &
   for a being Element of F holds multiplicity(p,a) <= 1;
   set Np = NormPolynomial p;
   deg p > 0 & deg p = len p - 1 by RATFUNC1:def 2,HURWITZ:def 2; then
   V: len p <> 0;
   U: p is non zero Element of the carrier of Polynom-Ring F
      by POLYNOM3:def 10; then
   W: Np = (LC p)" * p & LC p <> 0.F by RING_4:23; then
   (LC p) * Np = (LC p * (LC p)") * p by RING_4:11
              .= (1.F) * p by W,VECTSP_1:def 10 .= p; then
   consider c being non zero Element of F, q being Ppoly of F such that
   A: Np = c * q by AS,FIELD_8:9,FIELD_4:def 5;
   c = c * (1.F) .= c * (LC q) by RATFUNC1:def 7
    .= LC(c * q) by RING_5:5 .= 1.F by A,RATFUNC1:def 7; then
   reconsider Np as Ppoly of F by A;
   B: Roots Np = Roots p by V,POLYNOM5:61;
   now let a be Element of F;
     assume a is_a_root_of Np; then
     a is_a_root_of p by V,POLYNOM5:59; then
     multiplicity(p,a) >= 1 & multiplicity(p,a) <= 1 by AS,UPROOTS:52; then
     multiplicity(p,a) = 1 by XXREAL_0:1;
     hence multiplicity(Np,a) = 1 by ThsepsplB;
     end; then
   C: Np is Ppoly of F,(Roots Np) by FIELD_14:30;
   thus deg p = deg Np by U,REALALG3:11 .= card(Roots p) by B,C,RING_5:60;
   end;
hence thesis by A;
end;
