
theorem Th1:
  for p,q be Prime, k be Nat st
  k divides p*q holds k = 1 or k = p or k = q or k = p*q
proof
  let p,q be Prime, k be Nat;
  assume
A1: k divides p*q;
  per cases by PEPIN:2;
  suppose
    k,p are_coprime;
    then k divides q by A1,PEPIN:3;
    hence k = 1 or k = p or k = q or k = p*q by INT_2:def 4;
  end;
  suppose
    k gcd p = p;
    then p divides k by NAT_D:def 5;
    then consider l be Nat such that
A2: k = p*l by NAT_D:def 3;
    consider m be Nat such that
A3: k*m = p*q by A1,NAT_D:def 3;
    p*(l*m) = p*q by A2,A3;
    then l*m = q by XCMPLX_1:5;
    then l divides q by NAT_D:def 3;
    then l = 1 or l = q by INT_2:def 4;
    hence k = 1 or k = p or k = q or k = p*q by A2;
  end;
end;
