How does one prove Avogadro's law from Gay-Lussac's? To understand Gay-Lussac's law, Amedeo Avogadro said

In equal volumes of air under constant temperature and pressure, there are equal number of atoms (or molecules).

Let's take a hydrogen and an oxygen molecule. As the latter is larger in size, when stored in two separate equally-volumed containers, doesn't oxygen increase the pressure inside that container, if the same number of molecules occupied each container? (Does this make Avogadro's statement of constant pressure go wrong?) Again,
$$2mvN\cos\theta = F,$$
where $F$ is the force acting on the walls of the container, $N$ is the number of collisions of the molecules with the walls, $2mv$ is the change of momentum of the molecules, and $\theta$ is the angle of collision of the molecules on the walls.
Therefore, if Avogadro was correct, could anyone give me any reason why $N_\ce{H2} &gt; N_\ce{O2}$ or $v_\ce{H2} &gt; v_\ce{O2}?$
If Avogadro was correct, the above conditions must hold, as $m_\ce{H2} &lt; m_\ce{O2},$ right?
Could you please prove the above phenomena happen when Avogadro's statement above holds?
His name was Amedeo Avogadro, not Amidio nor Amadeo nor Avegadro nor Avagadro. The full name was Lorenzo Romano Amedeo Carlo Avogadro, Count of Quaregna and Cerreto.
Avogadro's law states that

equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.

More exactly, it also includes single atoms, e.g. for a case of noble gases, due the way how IUPAC defines a molecule.
But remember the Avogadro law implies ideal gases, similarly as the Gay-Lussac's law,Boyle's law and Charles' law. For ideal gases, sizes of molecules play no role. Both the size and volume of ideal gas molecules are considered negligible.
Real gases have deviations from this ideal behavior. See  van der Waals equation as the most used real gas state equation.

$$\left(p+a \frac{n^2}{V^2}\right)(V-n b)=nRT$$
$$\left(p+ \frac{a}{V_\mathrm{m}^2}\right)(V_\mathrm{m} - b)=RT$$

where parameters $a$ (addressing cohesive forces ) and $b$ (addresing non zero molecular volume) are specific for the particular gas.
Then the molar amount of a real gas for given $V$, $p$ and $T$ is the solution of the cubic equation for  variable $n$ :
$$pV - n ( pb + RT ) + a \frac{n^2}{V} - ab \frac{n^3}{V^2}  = 0 $$
It clearly shows that the same volumes of different real gases, at the same temperature and pressure, contain ( usually slightly ) different counts of molecules/atoms.