The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) :

Let $(X,\mathcal{U})$ be a uniform space such that for each member $U$ of some subbase for $\mathcal{U}$ there is a finite cover $A_{1}$, $A_{2}$,...,$A_{n}$ of $X$ such that $A_{i}\times A_{i}\subset U $ for each $i$. Then the space $(X,\mathcal{U})$ is totally bounded.

I was wondering how to prove this theorem. Is the proof of this theorem similar to Alexander's theorem? Moreover, if above theorem holds we can get a new proof of Tychonoff's theorem for completely regular spaces from following facts:

- A topology $\mathcal{T}$ for a set $X$ is the uniform topology for some uniformity for $X$ if and only if the topological space $(X,\mathcal{T})$ is completely regular.
- A uniform space is compact if and only if it is totally bounded and complete.
- The product of complete spaces is complete.
- The product of uniform spaces is totally bounded if and only if each coordinate space is totally bounded. (Only this fact needs to use above theorem)

Any and all help is appreciated.