Some techniques that were educational or helpful:

14.1.1:
They are jointly diagonalized by construction.

14.1.5:
Typo.  Last symbol should not have tilde.  As stated, a trivial bound is epsilon.

14.1.6:
You can get \delta instead of 3\delta by only bounding on of the two operators in the square.

14.1.8:
This is just a repeat of the classical derivation.

15.5.3:
Only the stated inequalities are used in the original proof.

15.5.4:
Add the typical subspace projector in (using the gental operator lemma) before applying the Hayashi-Nagaoka inequality.

17.3.1:
Entropy is now larger.  One interpretation is that there is now more state with which entanglement must be preserved.

17.4.1:
The equality is due to (17.42).