#### Mathematical English Usage - a Dictionary

##### by Jerzy Trzeciak

## on

The proof is by induction on n.

For brevity, we drop the subscript t on h_{t}.

Here H is a free R-module on as many generators as there are path components
of X.

the amount the gambler wins on the nth play

However, A clearly has a topology T on it, namely,......

On making use of the bound...... we conclude that......

This bound is integrable and on integration one again obtains the
same form of......

On restriction to B we have a homomorphism f:B→ C.

We establish our results both unconditionally and on the assumption
of the Riemann Hypothesis.

The following example shows that the degree of smoothness predicted by
Theorem 6 cannot be improved on.

We now show the following improvement on (2).

We now move on to the question of local normal forms.

Then (5) takes on the form...... [*Or*: takes the form]

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