reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem
  n is having_exactly_one_prime_divisor implies
  not n is having_at_least_three_different_prime_divisors
  proof
    given p being Prime such that
    p divides n and
A1: for r being Prime st r <> p holds not r divides n;
    given q1,q2,q3 being Prime such that
A2: q1,q2,q3 are_mutually_distinct and
A3: q1 divides n & q2 divides n and q3 divides n;
    q1 = p & q2 = p by A1,A3;
    hence contradiction by A2;
  end;
