This is a file I developed while writing a paper to make the life 
(of my imagination) a little bit easier. I stopped (perhaps, temporarily) 
to do it after someone point me out the guide due to Jerzy Trzeciak 
called "Writing mathematical papers in English". Trzeciak's book is 
done in a similar way to the following but it is much richer. In fact,
even names of some "chapters" turned out to be identical with Trzeciak's
ones (apart of fact that mine were written wrongly - with articles).

Feel free to use it and, please, feel obliged to contact me if you find 
a mistake in what follows. Thank you for any remarks and comments. 

PBN (pborod@math.uni.wroc.pl)


DEFINITIONS, NOTATION

A set .. is said to be .. if ... |  We let ... denote ...
We find it convenient to say that ... | Let us write .. for ..
... is called ... provided ... | We call ... a ... if ...
Let us say that ... is ... if ... | Write ... for ...
... is denoted by ... | From now on, let ... be ...
Define ... by setting ... | ... is defined as follows.
Set ... = ... | Put ... = ... | Fix ... = ... 
... is assumed to be ... | Recall that ... is ... if ...


PROOF - BEGINNING

Suppose that ... | Assume that ... | Suppose otherwise;
Assume for the contradiction that ... 
We may and shall assume that ... | We first show that ...
We first take the case of ... | Firstly, ...
We start by (letting) ... | We assert first that ...
We prove it as follows | The proof is based on the following observation
Let us first handle ... | Let us record the following result.
Let ... satisfy the condition ... | The construction proceeds as follows.
We argue by contradiction. | On the contrary |  Take any ... | Consider any ...
By ... we may suppose that ... | Let ... be ... provided by [Theorem].
We prove it by a method of ... | The proof will be analogous to that of ...



PROOF - CONTINUATION, DEDUCTION

If ..., then ... Otherwise ... | Whenever ..., then ...
Therefore, ... | Thus, ... | It then follows that ...
Hence, ... | Suppose further that ... | Also, ...
Moreover, ... | Furthermore, ... | By ..., we deduce ...
... implies ... | Consequently, ... | ... is ... due to ...
Next, ... | Accordingly, ... | To see this, observe that ...
Then, ... | Namely, ... | Indeed, ... | In addition, ...
... thanks to fact ... | Given that ..., then ...
Given ..., we have ... | To see this, ... 
Taking into account the above remark, we have ...
This gives [e.g. the equality] | In the same manner we see ...
Putting this together with ... we see that ...
... For suppose that ... | By ... we have ...
We also know ... | In particular, ... | At this point, note that ...
Notice that ... | We conclude that ... | It turns out that ...
One might treat ... as ...


PROOF - HOW TO HIDE A GAP.

It is clear that / It should be clear from the construction / Clearly, ...
It is routine to check that ... | It can be easily shown that ...
This follows by a result in [...]. | See [...] for further reading.
For the general discussion of .., see [...].
It is easy to observe that ... | We see at once that ...
We pass over the [trivial] case in which ...
The proof of ... are left for the reader. (!!)
... as the reader will easily see. | One can readily verify that ...
Obviously, ...


PROOF - END, CONTRADICTION

It suffices to show that ... | ... demonstrates ...
We can now complete the proof ... | ... and we are finished / done.
..., a contradiction. | ... is ..., as claimed.
This shows that ... | As ..., ... has the required property.
Finally, ... | ..., contradicting the assumption that ...
Thus, ... holds. | The proof is complete.



(MY) COMMON MISTAKES

you -> one
equal to -> equal
we have that -> we have
disjoint with -> disjoint from
equivalent with -> equivalent to
in the end, on the end -> at the end
thesis -> assertion
on step, in step -> at step