The approximate Brouwer Fixed Point Theorem (**aBFPT**) for the standard $n$-simplex is:

Let $f$ be a uniformly continuous function from $\Delta^n$ into itself. Then for each $\varepsilon>0$ there exists $x\in\Delta^n$ such that $d(f(x),x)<\varepsilon$.

Does anyone know a reference for a full proof of **aBFPT** via Sperner's Lemma in Bishop-style constructive mathematics (**BISH**)?

A constructive proof of **aBFPT** via Gale's Hex Theorem is in Hendtlass's PhD thesis. A constructive proof for the $2$-simplex is in this paper by van Dalen. That's all I could find.

If you don't know a reference but have a proof handy I'd very much like to see that as well.

Thank you in advance!

Note: The status of **BFPT** in constructive mathematics has been discussed in this MO post. For quick orientation: **aBFPT** may be viewed as constituting the constructive core of the full classical **BFPT**. The latter implies **LLPO** and is thus inherently nonconstructive (indeed, the two are equivalent over **BISH** since both are equivalent to **WKL** in the presence of countable choice). Classically, **BFPT** can be retrieved from **aBFPT** by a simple application of the Bolzano-Weierstrass theorem, which is equivalent to **LPO** over **BISH** and thus constructively inadmissible.