Homomorphism of External Direct Products

Theorem
Let:
 * $$\left({S_1 \times S_2, \circ}\right)$$ be the external direct product of two algebraic structures $$\left({S_1, \circ_1}\right)$$ and $$\left({S_2, \circ_2}\right)$$;
 * $$\left({T_1 \times T_2, *}\right)$$ be the external direct product of two algebraic structures $$\left({T_1, *_1}\right)$$ and $$\left({T_2, *_2}\right)$$.
 * $$\phi_1$$ be a homomorphism from $$\left({S_1, \circ_1}\right)$$ onto $$\left({T_1, *_1}\right)$$;
 * $$\phi_2$$ be a homomorphism from $$\left({S_2, \circ_2}\right)$$ onto $$\left({T_2, *_2}\right)$$.

Then the mapping $$\phi_1 \times \phi_2: \left({S_1 \times S_2, \circ}\right) \to \left({T_1 \times T_2, *}\right)$$ defined as:

$$\left({\phi_1 \times \phi_2}\right) \left({x, y}\right) = \left({\phi_1 \left({x}\right), \phi_2 \left({y}\right)}\right)$$

is a homomorphism from $$\left({S_1 \times S_2, \circ}\right)$$ to $$\left({T_1 \times T_2, *}\right)$$.