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

theorem
  34 = 2*17 & 34 has_exactly_two_different_prime_divisors
  proof
    thus
A1: 34 = 2*17;
    take P2, P17;
    thus P2 <> P17;
    thus P2 divides 34 by A1;
    thus P17 divides 34 by A1;
    let r be Prime such that
A2: r <> P2 & r <> P17;
    assume r divides 34;
    then r divides 2 or r divides 17 by A1,INT_5:7;
    hence thesis by A2,XPRIMES0:1,XPRIMES1:2,17;
  end;
