Corollary 4.3 and Lemma 4.2 below imply that and are essentially the same thing: randomly selecting the inputs of a universal EOM yields output prefixes whose probabilities are determined by the universal CEM.

*T* applies *EOM*_{0} to all
in dovetail fashion,
and simultaneously
simply reads randomly selected input bits forever.
At a given time, let string variable *z* denote *T*'s input
string read so far.
Starting at the right end of the unit interval [0,1),
as the
are being updated by the algorithm of
Theorem 4.1, *T*
keeps updating a chain of finitely many, variable, disjoint,
consecutive, adjacent, half-open intervals
*VI*(*x*)
of size
in alphabetic order on *x*,
such that *VI*(*y*) is to the right of *VI*(*x*) if .
After every variable update and each increase of
*z*, *T* replaces its output by the
*x* of the *VI*(*x*) with
.
Since neither *z* nor the
in the algorithm of Theorem 4.1
can decrease (that is, all interval boundaries can only shift left),
*T*'s output cannot either, and therefore is indeed EOM-computable.
Obviously the following holds:

and

Related links: In the beginning was the code! - Zuse's thesis - Life, the universe, and everything - Generalized Algorithmic Information - Speed Prior - The New AI