"We have" vs. "we have that" [closed]

Solution 1:

Both of these constructions are commonly used in mathematics. In ordinary (not mathematical) present-day English, these would be written as

... we have the fact that B = C.

Consider:

We have the fact that John is an idiot.
*We have that John is an idiot.
*We have John is an idiot.

Neither of the latter two constructions would be admissible in ordinary English, and I can find no evidence in Google books that they ever were admissible (although the middle one sounds marginally better to me, which means that if you had to choose one of these two, I would recommend that one). So I'd suggest rewriting it as one of the following:

Since A is true, we have the fact that B = C.
Since A is true, we learn that B = C.
Since A is true, we know (that) B = C.

(Or any of the other many ways of expressing this that are also grammatical in ordinary English.)

Note that I have dropped "the fact" from the first part of the sentence. Again, this is because it would not be grammatical in ordinary English:

*Since the fact the project was bungled, we know that John is an idiot.
Since the project was bungled, we know that John is an idiot.

Unlike the second part of the OP's original sentence, I don't believe it is even acceptable in mathematical English.

EDIT: I just realized that maybe you meant "since the fact A is true" to be parsed as "since (the fact A) is true" rather than "since the fact (A is true)". If that is the case, you can ignore the second part of my answer.