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

theorem Th27:
  2 <= n implies ex k be non zero Nat st
    Product primesFinS k <= n < Product primesFinS (k+1)
proof
  assume
A1: 2 <= n;
  defpred P[Nat] means n < Product primesFinS ($1+1);
  consider k be Nat such that
A2: 2 to_power k <= n & n < 2 to_power (k + 1) by A1,BINARI_6:2;
  2|^(k+1) <= Product primesFinS (k+1) by Th26;
  then n < Product primesFinS (k+1) by A2,XXREAL_0:2;
  then P[k];
  then
A3: ex k st P[k];
  consider m be Nat such that
A4: P[m] and
A5: for w be Nat st P[w] holds m <= w from NAT_1:sch 5(A3);
  m<>0 by A1,A4,NUMBER13:9;
  then reconsider m as non zero Nat;
  take m;
  reconsider m1=m-1 as Nat;
  Product primesFinS m <= n
  proof
    assume n < Product primesFinS m;
    then P[m1];
    then m1+1=m <=m1 by A5;
    hence thesis by NAT_1:13;
  end;
  hence thesis by A4;
end;
