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 $n$th play

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

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

On combining this with our other estimates in (3.5), we deduce that ......

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

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

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).

This is essentially a variation on the prime number theorem.

We will call on this version of the inverse theorem when we come to our applications in Section 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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