Poisson Counting Process

[kingwinner]

Let {N(t): t≥0} be a Poisson process of rate λ.
We are given that for a fixed t, N(t)=n.
Let Ti be the time of the ith event, i=1,2,...,n.

Then the event {T1≤t1, T2≤t2,...,Tn≤tn, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t1], (t1,t2],..., (tn-1,tn], and no events occur in (tn,t].
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I don't understand the 'exactly one' part.
For example, T2≤t2 just says that T2 is less than or equal to t2, and T2 can very possibly be less than t1 as well, right? (since it did NOT say that T2 MUST be larger than t1) In this case, we would then have more than one event occurring in [0,t1]. Why is this not allowed? I don't get it...

Can someone please explain? I would really appreciate it!

[EnumaElish]

Would N(t)=n be satisfied if T2 < t1?

[EnumaElish]

It seems to be a definitional choice that serves some purpose in the rest of the problem.

[kingwinner]

Would N(t)=n be satisfied if T2 < t1?

I think so! We can possibly have all n points in the interval [0,t1].

[kingwinner]

The book is trying to prove that the joint density function of T1,T2,...,Tn given that N(t)=n is given by (n!)/(t^n), 0<t1<t2<...<t_n<t

They are first trying to find the joint distribution function for 0<t1<t2<...<t_n<t.
i.e. P(T1≤t1,...Tn≤tn |N(t)=n)
and they commented that "the event {T1≤t1, T2≤t2,...,Tn≤tn, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t1], (t1,t2],..., (tn-1,tn], and no events occur in (tn,t". This is where I got totally confused...

Can someone please help?