theorem Th31:
  for n being non zero Nat holds
  LP<=6n+1(n), primenumber (1+primeindex LP<=6n+1(n)) are_not_twin
  proof
    let n be non zero Nat;
    set LP = LP<=6n+1(n);
    set s = primenumber(1+primeindex LP);
A1: LP <= s by Th28;
    s - LP <> 2 by Th30;
    hence thesis by A1,Th6;
  end;
