The circle and highlighted line confused me (logically, because I get the main idea) so I made a truth table, letting $P = x \notin A$ and $Q = x \in A ^\mathsf{c}$. Every truth assignment made sense except $P = F$, $Q = T$. If $P$ is false, is $x \in A$, and $Q$ is true, i.e., $x \in A ^\mathsf{c}$, how does that make the circled part true?
If $P$ ($x \not \in A$) is false and $Q$ ($x \in A^C$) is true, then the circled statement ('if $x \not \in A$ then $x \in A^C$') is true because the 'if' part is false and the 'then' part is true.... but you already knew that. So I think your real question is "How is it possible for $P$ to be false and $Q$ to be true (when in fact they are of course saying the same thing)?" Shouldn't the if ... then statement rule that out? And the answer is: No, the 'if $x \not \in A$ then $x \in A^C$' only rules out $x$ not being in $A$ ($P$ true) and $x$ also not being in $A^C$ ($Q$ false). It is the second statement 'if $x \in A$ then $x \not \in A^C$' that rules out the possibility of $x$ being in $a$ ($P$ false) and $x$ also being in $A^C$ ($Q$ true). So, between them, they do guarantee that $P$ and $Q$ say the same thing.
