Memoryless property for Poisson process

$$ P(N(20) = 6 \mid N(10) = 0) \ne P(N(20) = 6) $$ $$ P(N(20) = 6 \mid N(10) = 0) = P(N(20) -N(10) = 6) = P(N(10) = 6) = \frac{e^{-\lambda\cdot 10} (\lambda\cdot10)^6}{6!} $$ where $\lambda = \dfrac 1 {\text{4 seconds}} $.

It is not the Poisson distribution that is memoryless; it is the distribution of the waiting times in the Poisson process that is memoryless. And that is an exponential distribution.