reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th30:
  for n being non zero Nat holds
  (primenumber (1 + primeindex LP<=6n+1(n))) - LP<=6n+1(n) >= 4
  proof
    let n be non zero Nat;
    set LP = LP<=6n+1(n);
    set s = primenumber(1+primeindex LP);
    set r = LP;
    r in <=6n+1(n) /\ SetPrimes by XXREAL_2:def 8;
    then r in <=6n+1(n) by XBOOLE_0:def 4;
    then
A1: r <= 6*n+1 by Th7;
    s >= 6*n+5 by Th29;
    then s-r >= (6*n+5) - (6*n+1) by A1,XREAL_1:13;
    hence thesis;
  end;
