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

theorem KeyValue:
  for m being finite-support natural-valued ManySortedSet of SetPrimes,
      p being Prime st support m = {p} holds
    Product m = m.p
  proof
    let m be finite-support natural-valued ManySortedSet of SetPrimes,
        p be Prime;
    assume
A1: support m = {p};
      consider f being FinSequence of COMPLEX such that
A15:  Product m = Product f and
A16:  f = m * canFS (support m) by NAT_3:def 5;
Z1: m * <*p*> = <*m.p*>
    proof
D1:   rng <*p*> = {p} by FINSEQ_1:39;
d2:   p in SetPrimes by NEWTON:def 6;
      dom m = SetPrimes by PARTFUN1:def 2; then
B1:   dom (m * <*p*>) = dom <*p*> by D1,d2,RELAT_1:27,ZFMISC_1:31
          .= Seg 1 by FINSEQ_1:38;
B2:   dom (m * <*p*>) = dom <*m.p*> by B1,FINSEQ_1:38;
      for x being object st x in dom (m * <*p*>) holds
        (m * <*p*>).x = (<*m.p*>).x
      proof
        let x be object;
        assume
C1:     x in dom (m * <*p*>); then
C2:     x = 1 by B1,TARSKI:def 1,FINSEQ_1:2; then
        (m * <*p*>).x = m.(<*p*>.1) by FUNCT_1:12,C1
             .= (<*m.p*>).x by C2;
        hence thesis;
      end;
      hence thesis by B2,FUNCT_1:2;
    end;
    f = <*m.p*> by Z1,FINSEQ_1:94,A16,A1;
    hence thesis by RVSUM_1:95,A15;
  end;
