reserve m, n for Nat;

theorem Th16:
  for k being Nat, n being non zero Nat st
  support ppf n c= Seg (k+1) & not support ppf n c= Seg k holds k+1 is Prime
proof
  let k be Nat, n be non zero Nat;
  assume that
A1: support ppf n c= Seg (k+1) and
A2: not support ppf n c= Seg k;
  k+1 in support ppf n
  proof
    assume not k+1 in support ppf n;
    then
A3: {k+1} misses support ppf n by ZFMISC_1:50;
    support ppf n \ {k+1} c= Seg (k+1) \ {k+1} by A1,XBOOLE_1:33;
    then support ppf n c= Seg (k+1) \ {k+1} by A3,XBOOLE_1:83;
    hence thesis by A2,FINSEQ_1:10;
  end;
  then k+1 in support pfexp n by NAT_3:def 9;
  hence thesis by NAT_3:34;
end;
