1. The proof of ( 1 ) amounts to simple algebraic manipulation . 2. After considerable algebraic manipulation , this may be written as: 3. These equations are useful in reducing proofs about adjoint functors to algebraic manipulations . 4. It is much easier to perform algebraic manipulation on inequalities in this form. 5. Again, simple algebraic manipulation is required; for example, calorimetric experiments give: 6. For any fixed value of these identities can be obtained by tedious but completely straightforward algebraic manipulations . 7. Compared to Mongo, SM has a greatly enhanced command language, which includes algebraic manipulation of data vectors. 8. A few algebraic manipulations of your answer is all that is needed to arrive at the answer your program gave 9. Thus, with these three equations, it is possible to get the following result through some algebraic manipulation : 10. To be equal, the two sides have to be the same, either before or after algebraic manipulation .
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