## by

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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