Q:

Proof by contradiction - Getting my head around it

Hey there Math community!
I have a general question on contradiction and it's getting difficult to get my head around it.
Notes: 

I have some background in math and I have read several proofs by contradiction already
For the sake of the argument, let us assume the fundamental theorem of arithmetic

As per the general strategy for the proof, we assume the opposite of something that we wish to prove to begin with.
i.e A number that does NOT have a unique decomposition of primes.
We then proceed by a logical sequence of steps to show that this leads to a contradiction.
**Thus our original assumption was untenable and hence we have proved that all numbers have a unique decomposition of primes.
I have a problem understanding the star marked step.
It's like saying, if we want to prove the man is happy, let us assume the man is unhappy.
A logical sequence of steps leads to a contradiction.
Hence, our initial assumption is flawed, so the man is 'happy'?!?
What ensures that 'NOT unhappy' means 'happy' in the realm of math?
Thank you for your time :)

A:

This is called the law of excluded middle, and it is a kind of "meta-axiom" that most mathematicians accept.

