# Is it necessary to prove equality from both sides?

I have asked this question yesterday, and my friend told me, to rememeber to "prove it" also from the other side e.g. Let x $\in$ Conv($M+u$).....then $x$ $\in$ Conv($M$)+ $u$.

Why would somebody care for proving it from the other side, when using only equalities in the first place? If I had used something like

$X$ $\subseteq$ $B$ // one side

$B$ $\subseteq$ $X$ // other side

and from that conclude that $X = B$, I would agree. But why write out the other side when using only equalities?

I think that if using equalities, the other side is useless, am I right?

If not, please give me some examples.

• I don't think you need to re-hash through the whole process, but it would certainly be helpful to the reader if you at least pointed out the equality throughout and remark something like "We have shown that $x \in X \iff x \in B$ and therefore $X=B$." – White Shirt Mar 23 '16 at 0:02

As someone who grades proofs, if this were written in a beginning proof course I would probably take some points off. If this was done by a more experienced mathematician and I knew what they meant I wouldn't care. Any time I see a list of equalities or any list of consequences with justification I assume the argument something like this. $$P \implies P_1 \implies ...\implies Q$$ If you want to prove something like $P \iff Q$, with a similar list of consequences I think you should at least remark that each implication is actually an equivalence.
For an example of a statement proven with only equalities (where the converse isn't even true) consider something like "$x=0$ implies $xy=0$".