
theorem Th12:
  for p being Nat holds not p is prime iff
  p <= 1 or ex n being Element of NAT st n divides p & 1 < n & n < p
proof
  now
    let p be Nat;
    assume that
A1: not p is prime and
A2: not p<=1;
    consider n being Nat such that
A3: n divides p and
A4: n<>1 & n<>p by A1,A2,INT_2:def 4;
    reconsider n as Element of NAT by ORDINAL1:def 12;
    take n;
    thus n divides p by A3;
    thus 1<n & n<p by A2,A3,A4,Th7;
  end;
  hence thesis by INT_2:def 4;
end;
