# Note “that” or just Note? How may I improve this sentence?

I have the following sentence:

Note that by equation (1.1) for each point x belongs to set A there exists at least 2 points y in set B(x), and we need to make a measurable selection form it so that the set B(x) remains measurable.

First of all, should I use "note that" or just "note"? "for each apple" or just "each apple"? "exists" or "exist"?

Moreover, do I use "it" correcttly? By "it", I wish to refer to "at least 2 bananas"

Thank you!

• Is this a sentence that you wrote, or did you find it somewhere? I'm having trouble understanding what's meant by "by book". – stangdon Jul 1 '16 at 15:35
• @stangdon I wrote it, it is part of my thesis. Actually, "by book" is "by Theorem xxx" in my thesis, and "apple", "banana", "basket" are all some long terminolagys... you may replace "by book" by "by some references"... – JumpJump Jul 1 '16 at 15:38
• OK, I understand. We wouldn't say "by book" there - I think the phrase you want is something more like "according to Theorem XXX" or "according to some references". – stangdon Jul 1 '16 at 15:43
• @stangdon Thx! I just want to somehow write my question here without writing too many wired terminolage... – JumpJump Jul 1 '16 at 15:45
• I understand, but it might actually help to post the exact sentence we're using, because it might make a difference to the answer to your question. – stangdon Jul 1 '16 at 17:03

Note that by equation (1.1)
observe that by using equation (1.1)

might better be expressed using the math-speak syntax of

Note that from equation (1.1)
From equation (1.1)
Using equation (1.1)
Given equation (1.1)