Multi-track Turing machine

A Multitrack Turing machine is a specific type of multi-tape Turing machine. 
In a standard n-tape Turing machine, n heads move independently along n tracks. In a n-track Turing machine, one head reads and writes on all tracks simultaneously. A tape position in a n-track Turing Machine contains n symbols from the tape alphabet. It is equivalent to the standard Turing machine and therefore accepts precisely the recursively enumerable languages.
Formal definition.
A multitrack Turing machine with formula_1-tapes can be formally defined as a 6-tuple formula_2, where
A non-deterministic variant can be defined by replacing the transition function formula_11 by a "transition relation" formula_12.
Proof of equivalency to standard Turing machine.
This will prove that a two-track Turing machine is equivalent to a standard Turing machine. This can be generalized to a n-track Turing machine. Let L be a recursively enumerable language. Let M= formula_13 be standard Turing machine that accepts L. Let M' is a two-track Turing machine. To prove M=M' it must be shown that M formula_14 M' and M' formula_14 M
If all but the first track is ignored then M and M' are clearly equivalent. 
The tape alphabet of a one-track Turing machine equivalent to a two-track Turing machine consists of an ordered pair. The input symbol a of a Turing machine M' can be identified as an ordered pair [x,y] of Turing machine M. The one-track Turing machine is:
M= formula_18 with the transition function formula_19
This machine also accepts L.