d to
justify the corresponding step of the graphical transformation.

The axioms are just four in number,
divided into two types, as follows:
the "arithmetic initials" I1 and I2,
the "algebraic  initials" J1 and J2.

(I use a trivially different J1 here.)

Notice that all of the following axioms are equations.
Thus, the inference steps that they permit are always
reversible.  In the proof annotation scheme below,
I will use a double bar "===" to mark this fact,
and I leave it to the reader to decide in whic