Understanding the Poisson Memorylessness Proof

Question

I was hoping someone can explain to me step by step the proof of Poisson memorylessness property. First i understand that ,

1. Memoryless: P(X>s + t|X>t) = P(X>s).

From that point the rule of conditional probability is used which will then simplify to :

2.P(X>s + t, X > t)/P(X>t)

But from this point, i don't understand the mathematics involved in these subsequent phases. Please explain as you would to a novice how each step is simplified, my background in mathematics is average.

3.P(X>s + t)/P(X>t)
4.e−λ(s+t)/e−λt
5.e−λs
6.P(X>s)

Answer 1

3. $[X\gt s+t,X\gt t]=[X\gt s+t]$ since $s\geqslant0$.

4. Definition of exponential CDF is $P(X\gt x)=\mathrm e^{-\lambda x}$ for every $x\geqslant0$.

5. Obvious.

6. Definition of exponential CDF.