 reserve n,i for Nat;
 reserve p for Prime;

theorem Matsu1:
  for f being bag of SetPrimes st f.p = p |^ n holds
    p |^ n divides Product f
  proof
    let f be bag of SetPrimes;
    assume
AA: f.p = p |^ n;
    consider g being FinSequence of NAT such that
A2: Product f = Product g & g = f*canFS(support f) by Matsu0;
    reconsider PP = Product g as Nat;
    p in SetPrimes by NEWTON:def 6; then
B0: p in dom f by PARTFUN1:def 2;
B1: p in support f by AA,PRE_POLY:def 7;
    rng canFS support f = support f by FUNCT_2:def 3; then
    consider y being object such that
B2: y in dom canFS support f & p = (canFS support f).y by B1,FUNCT_1:def 3;
B3: y in dom g by A2,FUNCT_1:11,B2,B0;
    f.p = g.y by A2,B2,FUNCT_1:13; then
    f.p in rng g by FUNCT_1:3,B3;
    hence thesis by A2,AA,NAT_3:7;
  end;
