
theorem Th25:
  for n,f,d,n1,a,q being Element of NAT st n-1 = q|^n1*d & q|^n1 > d &
  d > 0 & q is prime & a|^(n-'1) mod n = 1 & (a|^((n-'1) div q)-'1) gcd n = 1
  holds n is prime
proof
  let n,f,d,n1,a,q be Element of NAT;
  assume that
A1: n-1=q|^n1*d & q|^n1>d & d>0 and
A2: q is prime;
  set f=q|^n1;
  assume
A3: a|^(n-'1) mod n = 1;
  assume
A4: (a|^((n-'1) div q)-'1) gcd n = 1;
  for q1 being Element of NAT st q1 divides f & q1 is prime holds ex a
being Element of NAT st a|^(n-'1) mod n = 1 & (a|^((n-'1) div q1)-'1) gcd n = 1
  proof
    let q1 be Element of NAT;
    assume
A5: q1 divides f;
    assume
A6: q1 is prime;
    consider a be Element of NAT such that
A7: a|^(n-'1) mod n = 1 & (a|^((n-'1) div q)-'1) gcd n = 1 by A3,A4;
    take a;
    thus thesis by A2,A5,A6,A7,Th18;
  end;
  hence thesis by A1,Th24;
end;
