reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th47:
  for n being non zero Nat, i being Nat st i in dom PrimeDivisorsFS(n) holds
  PrimeDivisorsFS(n).i is prime
  proof
    let n be non zero Nat;
    set f = PrimeDivisorsFS(n);
    let i be Nat;
    assume i in dom f;
    then f.i in rng f by FUNCT_1:def 3;
    then f.i in PrimeDivisors(n) by FINSEQ_1:def 14;
    then ex p being Prime st p = f.i & p divides n;
    hence thesis;
  end;
