Gauss Theorem (The fundamental theorem of arithmetics).

     Every natural number can be factored into prime factors in the only way.

     Proof. Let's assume, that some number A can be factored into prime factors in two ways:

A = p₁ p₂ ... p_n = q₁ q₂ ... q_m .

     Using the corollary of Gauss Theorem, we get, that one of the numbers q_i must coincide with p_i. Not breaking the community, it can be considered, that it is q₁. Reducing by p₁, we come to the equality

p₂ ... p_n = q₂ ... q_m .

     Repeating this reasoning sufficient quantity times, we finelly come to a conclusion, that n = m and the multipliers in both factorizations are equal, what was to be proved.