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
  35 = 5*7 & 35 has_exactly_two_different_prime_divisors
  proof
    thus
A1: 35 = 5*7;
    take P5, P7;
    thus P5 <> P7;
    thus P5 divides 35 by A1;
    thus P7 divides 35 by A1;
    let r be Prime such that
A2: r <> P5 & r <> P7;
    assume r divides 35;
    then r divides 5 or r divides 7 by A1,INT_5:7;
    hence thesis by A2,XPRIMES0:1,XPRIMES1:5,7;
  end;
