By continuity $\langle$By the continuity$\rangle$ of $f$, ......
By assumption, ......
By the above, ......
By the induction hypothesis, ......
By and large, we shall use the same notation as in ...... [= In general]
term-by-term differentiation
a 3 by 4 matrix = a $3\times 4$ matrix
Thus we need only alter our constants by a factor of 2 to deal with this case.
By quasi-equation we understand a sentence of the form ......
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