
theorem
  for f being increasing Arithmetic_Progression st
    for i being Nat holds f.i is Prime holds
      difference f is even Nat
  proof
    let f be increasing Arithmetic_Progression;
    assume
A1: for i being Nat holds f.i is Prime;
    f.2 > f.1 by SEQM_3:1; then
    f.2 > 2 by A1,FirstAPPrime,XXREAL_0:2; then
    reconsider f2 = f.2, f1 = f.1 as 2_greater Prime
      by NAT_6:def 1,A1,FirstAPPrime;
C5: f2 - f1 > 0 by SEQM_3:1,XREAL_1:50;
    f.(1+1) - f.1 = f.1 - f.0 by NUMBER06:5;
    hence thesis by C5,NUMBER06:def 5;
  end;
