 reserve n,i for Nat;

theorem Matsu:
  for f being bag of SetPrimes holds Product f <> 0
  proof
    let f be bag of SetPrimes;
    consider g being FinSequence of COMPLEX such that
A2: Product f = Product g & g = f*canFS(support f) by NAT_3:def 5;
    assume Product f = 0; then
    consider k be Nat such that
H1: k in dom g & g.k = 0 by A2,RVSUM_1:103;
h1: dom g c= dom canFS support f by A2,RELAT_1:25; then
H3: f.((canFS support f).k) = 0 by A2,H1,FUNCT_1:13;
    (canFS support f).k in rng canFS support f by FUNCT_1:3,H1,h1; then
    (canFS support f).k in support f by FUNCT_2:def 3;
    hence thesis by H3,PRE_POLY:def 7;
  end;
