 reserve n for Nat;
 reserve F for Field,
         p for irreducible Element of the carrier of Polynom-Ring F,
         f for Element of the carrier of Polynom-Ring F,
         a for Element of F;

theorem
   f is non constant implies ex p being irreducible Element of
   the carrier of Polynom-Ring F st emb(f,p) is with_roots
   proof
     assume f is non constant; then
     consider p being Element of the carrier of Polynom-Ring F such that
A2:   p is_a_irreducible_factor_of f by Th4;
     reconsider p as irreducible Element of the carrier of Polynom-Ring F
       by A2;
     take p;
     consider q being Element of the carrier of Polynom-Ring F such that
A3:   p * q = f by A2,GCD_1:def 1;
A4:   emb(f,p) = (PolyHom (emb p)).p * (PolyHom (emb p)).q by A3,GROUP_6:def 6
     .= emb(p,p) *' emb(q,p) by POLYNOM3:def 10;
     (KrRoot p) is_a_root_of emb(p,p) by Th43; then
     emb(p,p) is with_roots by POLYNOM5:def 8;
     hence thesis by A4;
   end;
