
theorem ThSepPR:
for F being Field
for p,q being non constant Element of the carrier of Polynom-Ring F
for r being Element of the carrier of Polynom-Ring F
st p = q *' r holds p is separable implies q is separable
proof
let F be Field, p,q be non constant Element of the carrier of Polynom-Ring F;
let r be Element of the carrier of Polynom-Ring F;
assume A: p = q *' r;
assume B: p is separable;
reconsider r as non zero Polynomial of F by A;
now assume q is inseparable; then
  consider E being FieldExtension of F such that
  D: not for a being Element of E holds multiplicity(q,a) <= 1 by ThSep02;
  consider a being Element of E such that
  E: multiplicity(q,a) > 1 by D;
  F: multiplicity(p,a) = multiplicity(q,a) + multiplicity(r,a) by A,UP55;
  multiplicity(q,a) + multiplicity(r,a) >= multiplicity(q,a) by NAT_1:11;
  then multiplicity(p,a) > 1 by F,E,XXREAL_0:2;
  hence contradiction by B,ThSep02;
  end;
hence thesis;
end;
