Multiplication Theorem on Probability and proof

[alfred2]

Hi Everyone!

I'm with Conditional Probability and I don't understan this theorem.

Theorem:
If
http://imageshack.us/a/img28/1349/1qg.png
then
http://imageshack.us/a/img209/8829/aor.png
Proof:
All the conditional probabilities are well defined, since
http://imageshack.us/a/img197/8938/4xp.png
We can rewrite the right site of the equality as follows
http://imageshack.us/a/img62/3095/5d9.png
Obviously we can simplify the terms through
http://imageshack.us/a/img855/7523/ikm.png
Can anyone say me how does the simplification work? And why it is so important to be sure that

Thank you =)

 

[I like Serena]

Welcome to MHB, alfred! :)

The simplification is a consequence of how fractions are multiplied and simplified in general.
Consider for instance:
$$\frac 3 4 \cdot \frac 4 5 \cdot \frac 5 6 = \frac 3 {\cancel 4} \cdot \frac {\cancel 4} {\cancel 5} \cdot \frac {\cancel 5} 6 = \frac 3 6$$

 

[alfred2]

Thanks! You were right ;)