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
  36 = 2*2*3*3 & 36 has_exactly_two_different_prime_divisors
  proof
    thus
A1: 36 = 2*2*3*3;
    take P2, P3;
    thus P2 <> P3;
    2*2*3*3 = 2*(2*3*3);
    hence P2 divides 36;
    thus P3 divides 36 by A1;
    let r be Prime such that
A2: r <> P2 & r <> P3;
    assume r divides 36;
    then r = 1 or r = 2 or r = 3 or r = 4 or r = 6 or r = 9 or
    r = 12 or r = 18 or r = 36 by Th15;
    then r = 2 or r = 3 by INT_2:def 4,XPRIMES0:4,6,9,12,18,36;
    hence thesis by A2;
  end;
