, 
, 
and 
be defined as follows.
The math problem we address is:
(HGRAIL)
Let A, 
, 
, 
, 
be matrices of compatible size
be given.
Solve 
,
, 
, 
, and 
for a, b, c and for
, 
, 
and 
. When can they be solved?
If these equations can be solved, find formulas for the solution.
We shall make the strong assumption that each 
is invertible.
While this assumption
turns out to be valid, making it at this point is cheating.
Ironically, we recommend strongly that the user make heavy
invertibility assumptions at the outset of a session.
Later, after the main ideas have been discovered,
the user can selectively relax them and thereby obtain
more general results.