Typo in I),1): It should be Az=0, not Ax=0.
Style in II),1): use z instead of x for consistency of notation between I and II
I) n = rank + nullity
1) x = y + z, Ay ≠ 0, Ax = 0, n = r’ + ν
2) Ax = Ay
3) Ay = col space of A, dim r (rank)
4) y = sol space of A, dim r’
5) y1 ≠ y2 → Ay1 ≠ Ay2 → y1 ≠ y2 → r’ = r
6) n = r + ν
II) row rank = column rank
1) Ax = 0 → x is null space of cols (def) and rows (a_{li}x_{i}=0) of A
2) n = r + ν in either case
3) row rank = column rank
The intent is to convey the point of the proof as logically, intuitively, and conciseley as possible, to be easily recalled. Individual steps can then be filled in. Shortcuts should be obvious. It is assumed one has seen standard text-book proofs. Note convention sum over repeated indices.