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 Th29:
  for n being non zero Nat holds
  primenumber (1 + primeindex LP<=6n+1(n)) >= 6*n+5
  proof
    let n be non zero Nat;
    set M = LP<=6n+1(n);
    set N = 1+primeindex M;
    assume primenumber(N) < 6*n+5;
    then
A1: primenumber(N) < 6*n+4+1;
A2: n >= 0+1 by NAT_1:13;
    then
A3: 3*n >= 3*1 by XREAL_1:66;
    then 3*n+1 >= 3+1 by XREAL_1:6;
    then
A4: 6*n+2 <> 2;
    2 divides 2*(3*n+1);
    then
A5: not 6*n+2 is prime by A4;
    2*n >= 2*1 by A2,XREAL_1:66;
    then 2*n+1 >= 2+1 by XREAL_1:6;
    then
A6: 6*n+3 <> 3;
    3 divides 3*(2*n+1);
    then
A7: not 6*n+3 is prime by A6;
    3*n+2 >= 3+2 by A3,XREAL_1:6;
    then 2*(3*n+2) >= 2*5 by XREAL_1:64;
    then
A8: 6*n+4 <> 2;
    2 divides 2*(3*n+2);
    then
A9: not 6*n+4 is prime by A8;
A10: primenumber(N) in SetPrimes by NEWTON:def 6;
    primenumber(N) <= 6*n+3+1 by A1,NAT_1:13;
    then primenumber(N) < 6*n+3+1 by A9,XXREAL_0:1;
    then primenumber(N) <= 6*n+3 by NAT_1:13;
    then primenumber(N) < 6*n+2+1 by A7,XXREAL_0:1;
    then primenumber(N) <= 6*n+1+1 by NAT_1:13;
    then primenumber(N) < 6*n+1+1 by A5,XXREAL_0:1;
    then primenumber(N) <= 6*n+1 by NAT_1:13;
    then primenumber(N) in <=6n+1(n);
    then primenumber(N) in <=6n+1(n) /\ SetPrimes by A10,XBOOLE_0:def 4;
    then primenumber(N) <= M by XXREAL_2:def 8;
    hence thesis by Th28;
  end;
