How the product of two integrals is iterated integral? $\int\cdot \int = \iint$

Note first, that for any integrable $f \colon \def\R{\mathbb R}\R \to \R$ and any $\alpha \in \R$ we have $$ \int_\R \alpha f(x)\, dx = \alpha \int_\R f(x) \, dx $$ Now note that with $\alpha := \int_\R f(y)\, dy$ (this depends on $f$, but given $f$ it is a constant) this gives $$ \int_\R \left(\int_\R f(y)\, dy\right)\, f(x)\, dx = \int_\R f(y)\, dy \cdot \int_\R f(x)\, dx $$ Now, for every fixed $x \in \R$, we apply the above again, now for $\alpha = f(x)$ (which is not depending on $y$), giving $$ \left(\int_\R f(y)\, dy\right)\, f(x) = \int_\R f(x)f(y)\, dy $$ altogether $$ \int_\R \int_\R f(y)f(x)\, dy\, dx = \int_\R f(y)\, dy \cdot \int_\R f(x)\, dx $$