\Rightarrow "Let

be an equivalence relation on

. For any

either

or

"

My professor gave the following proof to the preceding lemma in his notes, but he starts it by saying "It is enough to assume" then footnoting this and writing "no it isn't". So how do I need to add to this to make it a complete proof?

It is enough to assume (footnoted here)

and deduce that

First, fix

For any

,

means

(similarly)

(symmetry)

Therefore

(transitivity), i.e.,

That is, for every

In other words,

In particular,

and also

Therefore

So, if

choose some

Then

and

, so