Why Kinetic energy of Gas molecules independent of Mass?

By Kinetic Theory of Gas,

$K.E = frac32 RT$ (i.e it is independent of mass of the gas)

Its proof is as follows:

We know , $P = frac13 Dcdot v^2$ (where $D$ is mass density and $v$ is average of the squared velocity of molecules)

Multiply both sides by $V$(Volume)
$$
PV = frac13 Mv^2
$$
Multiply and divide by $2$ in the rhs of the equation
$$
PV= frac23 × frac12 Mv²
$$
$$
PV= frac23 × K.E ~~~~~~~~[1]
$$
$$
RT= frac23 × K.E
$$
$$
K.E = frac32 RT
$$

But in [$1$] we used that $frac12 Mv^2= K.E$ (i.e $K.E$ as a function of Mass).
In the end we got $K.E= frac32 RT$ (i.e $K.E$ is independent of Mass)

Please explain how $K.E$ of gases is independent of Mass