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

theorem Th30:
  for p be Prime, k be Nat st k <= primeindex p
    holds p,Product primesFinS k are_coprime
proof
  let p be Prime, k be Nat;
  assume k <= primeindex p;
  then
A1: primenumber k <= p by NUMBER13:12;
  defpred P[Nat] means $1 <=k implies p,Product primesFinS ($1) are_coprime;
  Product primesFinS 0 = 1 by RVSUM_1:94;
  then
A2: P[0] by WSIERP_1:9;
A3: P[n] implies P[n+1]
  proof
    assume
A4:   P[n];
    set n1=n+1;
    assume n1 <= k;
    then
A5:   n < k by NAT_1:13;
    then
A6:   p,Product primesFinS n are_coprime by A4;
A7:   Product primesFinS (n+1) = Product (primesFinS n) * (primenumber n)
    by Th25;
    primenumber n < primenumber k by A5,MOEBIUS2:21;
    then primenumber n < p by A1,XXREAL_0:2;
    then primenumber n,p are_coprime by NUMBER06:3;
    then p,Product primesFinS (n+1) are_coprime by A6,A7,INT_2:26;
    hence thesis;
  end;
  P[n] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
