Answer
See the solution.
Work Step by Step
To prove this, we will prove the two conditional sentences
'$m^2=n^2 \rightarrow (m=n \lor m=-n)$' and
'$(m=n \lor m=-n) \rightarrow m^2=n^2$'.
$Proof.$
To prove '$m^2=n^2 \rightarrow (m=n \lor m=-n)$', we will use a direct proof. So suppose $m^2=n^2$. We subtract $n^2$ from both sides to get $m^2-n^2=0$. Then factoring the left hand side gives $(m-n)(m+n)=0$. This means either $m-n=0$ or $m+n=0$. Therefore, either $m=n$ or $m=-n$. Thus, we have
shown '$m^2=n^2 \rightarrow (m=n \lor m=-n)$'.
Now, to prove '$(m=n \lor m=-n) \rightarrow m^2=n^2$', we will prove the equivalent statement '$(m=n \rightarrow m^2=n^2) \land (m=-n \rightarrow m^2=n^2)$'. So, first, suppose $m=n$. Then by substitution, we have $m^2=mm=nn=n^2$. Thus '$(m=n \rightarrow m^2=n^2)$'. Next, suppose $m=-n$. Then by substitution, $m^2=mm=(-n)(-n)=n^2$.
Hence '$(m=-n \rightarrow m^2=n^2)$'. Thus we have shown
'$(m=n \rightarrow m^2=n^2) \land (m=-n \rightarrow m^2=n^2)$', which is equivalent to '$(m=n \lor m=-n) \rightarrow m^2=n^2$'.
Finally, since '$m^2=n^2 \rightarrow (m=n \lor m=-n)$' and
'$(m=n \lor m=-n) \rightarrow m^2=n^2$', we have
'$m^2=n^2 \leftrightarrow(m=n \lor m=-n)$'.$_\Box$
