I'm struggling to read this proof, since I'm used to read implications and not english words. Are these implications in this proof correctly assigned ?
What does in this setting "necessity of condition mean?"
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
The main ambiguity is what "the condition" is.
From the context, it is clear that "the condition" is the second, more complicated thing in the theorem, which is meant to characterize the simpler first half (when the polytope is integral.) The author could have made this clearer, but it makes sense if you fit it with the explanation that follows.
With this interpretation, everything is fine. 'integral $\implies$ "the condition" ' is the same as saying "the condition" is necessary, and '"the condition" $\implies$ integral' is the same as saying "the condition" is also sufficient.
Personally, I find the tradition of using the terms "necessary" and "sufficient" more trouble than they're worth, for exactly the reasons encountered here: the mental overhead required to parse it.
That is a somewhat old style way of saying "if", "only-if" parts.
A theorem of the form "$P$ if and only if $Q$" is equivalent to "In order for $P$ to hold, a necessary and sufficient condition is $Q$". So to prove the necessity of $Q$ for $P$ is to prove "$P$ implies $Q$"; to prove the sufficiency of $Q$ for $P$ is to prove "$Q$ implies $P$".
The author uses "iff" in his statement of the theorem but employs the terms "sufficiency" and "necessity" would cause confusion. Fortunately we can infer his meaning within the context.