Hint $\ $ When debugging proofs on abstract objects, the problem may become simpler to spot after specializing to more concrete objects. The symbols $\rm\:x,y,z\:$ denote abstract numbers, so let's specialize them to their concrete number values: $\rm\:x = 5,\: y=7,\: z = 6,\:$ yielding this "proof"
5 + 7 &=&\: 2\cdot 6 \\
5- 2\cdot 6 &=&\: -7 \\
\cdots\ &=&\ \cdots \\
(5-6)^2\! &=&\: (7-6)^2 \\
5-6\ \ \:&=&\:\ \ 7-6\:
Now you can determine which inference is incorrect by determining the first false equation above. If equation number $\rm\: n\!+\!1\:$ is false then the inference from equation $\rm\:n\:$ to $\rm\:n\!+\!1\:$ is incorrect. You'll find that the final inference is false, the last equation being false. Now the error is clear.
Analogous methods prove helpful generally: when studying abstract objects and something is not clear, look at concrete specializations to gain further insight on the general case.