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

theorem Th32:
  for n,m being Nat st
    for k being Nat st k < m holds primenumber k divides n
  holds Product primesFinS m divides n
proof
  let n,m be Nat such that
A1: for k being Nat st k < m holds primenumber k divides n;
  defpred P[Nat] means $1 <= m implies Product primesFinS $1 divides n;
  Product primesFinS 0 = 1 by RVSUM_1:94;
  then
A2: P[0] by INT_2:12;
A3: for i be Nat st P[i] holds P[i+1]
  proof
    let i such that
A4:   P[i];
    assume i+1 <=m;
    then i < m by NAT_1:13;
    then
A5:   Product primesFinS i divides n & primenumber i divides n by A1,A4;
    i in NAT by ORDINAL1:def 12;
    then primeindex primenumber i = i by NUMBER10:def 4;
    then primenumber i, Product primesFinS i are_coprime by Th30;
    then (primenumber i) * Product primesFinS i divides n by A5,PEPIN:4;
    then Product primesFinS (i+1) divides n by Th25;
    hence thesis;
  end;
  for i be Nat holds P[i] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
