 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem PrimesSet2:
  n+1 is Prime implies SetPrimes (n+1) = SetPrimes n \/ {n+1}
  proof
A1: SetPrimes (n+1) = SetPrimes /\ (Seg n \/ {n+1}) by FINSEQ_1:9
      .= (SetPrimes n) \/ (SetPrimes /\ {n+1}) by XBOOLE_1:23;
    assume n+1 is Prime; then
    n+1 in SetPrimes by NEWTON:def 6;
    hence thesis by A1,ZFMISC_1:46;
  end;
