I'm currently trying to understand the proof of the (elementary) JSJ-Decomposition in [Allen Hatcher's notes on 3-Manifolds](http://www.math.cornell.edu/%7Ehatcher/3M/3Mdownloads.html).


Specifically, I have trouble understanding the application of proposition 1.7 to corollary 1.8.


Proposition 1.7 states:



> 
> For a compact irreducible 3-Manifold M there is a bound on the number of
> components in a system $S = S\_1 \cup \dots \cup S\_n$ of disjoint closed incompressible surfaces $S\_i \subset M$ such that no component of $M | S$ ("M split along S") is a product $T \times I$ with $T$ a closed surface.
> 
> 
> 


Corollary 1.8 states:



> 
> In a compact connected irreducible 3-Manifold M there exists a (possibly empty) collection T of disjoint incompressible tori, such that each component of $M|T$ is atoroidal.
> 
> 
> 


In the proof, one starts with $T=\emptyset$, and then applies the following steps repeatedly:


1. If $M|T$ is atoroidal, we are done.
2. Otherwise, there exists an incompressible torus in $M|T$ which is not boundary-parallel (by definition of atoroidal). Add this torus to T. Go to step 1.


To show that this procedure will finish in a finite number of steps, the following argument is given:  

Assume otherwise. Then there exist arbitrarily large collections T of disjoint incompressible tori, **such that no component of $M|T$ is a product of $I$ and a torus, in contradiction to proposition 1.7**.


I don't see why proposition 1.7 applies here - the condition in 1.7 is stronger, namely that no component of $M|T$ is a product of $I$ and a closed surface.


So my question is: why and how is prop. 1.7 applicable here?


I hope this question is suitable for MathOverflow. If not, a hint where I might get an answer would be nice.



JSJ-Decomposition - proof of finiteness in Hatcher's Notes 
Since we can only do JsJ-decomposition for toroidal manifold, as you say, we can assume that manifold M is toroidal. cutting M along the essential torus, the induced manifold or manifolds 
are both not the surface I-bundle and possible toroidal. We can do Jsj-decomposition again and again until that each component is atoroidal. And off course, each component is not surface I-bundle ,since we cut manifold along the essential torus. 


Prop 1.7 implies that the maximal time for jsj-decomposition is finite.That implies 1.8. 


