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

theorem Th25:
  Product primesFinS (n+1) = Product (primesFinS n) * (primenumber n)
proof
  len primesFinS (n+1) = (n+1) by NUMBER13:def 1;
  then
A1: primesFinS (n+1) =
  ((primesFinS (n+1)) | n) ^ <*(primesFinS (n+1)).(n+1)*>
  by FINSEQ_3:55;
A2:  n < n+1 by NAT_1:13;
  then
A3: ((primesFinS (n+1)) | n) = primesFinS n by NUMBER13:17;
  (primesFinS (n+1)).(n+1) = primenumber n by A2,NUMBER13:def 1;
  then primesFinS (n+1) = (primesFinS n) ^ <*primenumber n*> by A1,A3;
  then Product primesFinS (n+1) = Product (primesFinS n) * (primenumber n)
    by RVSUM_1:96;
  hence thesis;
end;
