 reserve n for Nat;

theorem Th4:
  for R being non almost_left_invertible factorial domRing,
      a being non zero NonUnit of R
   ex b being Element of R st b is_a_irreducible_factor_of a
    proof
      let R be non almost_left_invertible factorial domRing,
      a be non zero NonUnit of R;
      consider F being non empty FinSequence of R such that
A1:    F is_a_factorization_of a by RING_2:def 12;
      len F <> 0; then
      consider G being FinSequence of R, d being Element of R such that
A2:    F = G ^ <*d*> by FINSEQ_2:19;
      take d;
      Product(G) * Product(<*d*>) = a by A1,A2,GROUP_4:5;then
A3:    Product(<*d*>) divides a by GCD_1:def 1;
      reconsider lf = len F as Element of dom F by FINSEQ_5:6;
      len F = len G + len<*d*> by A2,FINSEQ_1:22
           .= len G + 1 by FINSEQ_1:39; then
      F.lf = d by A2,FINSEQ_1:42;
      hence thesis by A1,A3,GROUP_4:9;
end;
