
theorem
for F being Field
for p being non zero Element of the carrier of Polynom-Ring F
holds (ex a being Element of F st multiplicity(p,a) > 1)
      iff p gcd (Deriv F).p is with_roots
proof
let F be Field, p be non zero Element of the carrier of Polynom-Ring F;
set g = p gcd (Deriv F).p;
A: now assume ex a being Element of F st multiplicity(p,a) > 1; then
   consider a being Element of F such that A1: multiplicity(p,a) > 1;
   A2: a is_a_root_of p by A1,UPROOTS:52;
       eval(p,a) = 0.F by POLYNOM5:def 7,A1,UPROOTS:52; then
   A3: rpoly(1,a) divides p by RING_5:11;
       eval((Deriv F).p,a) = 0.F by A1,A2,multi4; then
       rpoly(1,a) divides (Deriv F).p by RING_5:11; then
       eval(g,a) = 0.F by A3,RING_4:52,RING_5:11;
   hence g is with_roots by POLYNOM5:def 7;
   end;
now assume g is with_roots; then
   consider a being Element of F such that
   B7: a is_a_root_of g;
       a is_a_root_of p by B7,mm1,RING_4:52; then
   B4: multiplicity(p,a) > 1 iff eval((Deriv F).p,a) = 0.F by multi4;
       a is_a_root_of (Deriv F).p by B7,mm1,RING_4:52;
   hence ex a being Element of F st multiplicity(p,a) > 1 by B4;
   end;
hence thesis by A;
end;
