reserve a,b,d,n,k,i,j,x,s for Nat;

theorem Th21:
  for n be non zero Nat, p be Prime st support pfexp n = {p} holds
     n = p |^ (pfexp n).p
proof
  let n be non zero Nat, p be Prime such that
A1: support pfexp n = {p};
  set b = ppf n;
  consider f be FinSequence of COMPLEX such that
A2: Product b = Product f and
A3: f = b*canFS(support b) by NAT_3:def 5;
A4: support pfexp n = support b by NAT_3:def 9;
  then
A5: p in support b c= dom b by A1,PRE_POLY:37,TARSKI:def 1;
  f = b*<*p*> by A3,A1,A4,FINSEQ_1:94
  .= <* b.p *> by A5,FINSEQ_2:34;
  then Product b = ::b.p by A2
   p |^ (p |-count n) by A2,A5,A4,NAT_3:def 9;
  then n = p |^ (p |-count n) by NAT_3:61;
  hence thesis by NAT_3:def 8;
end;
