I'm reading a paper that's trying to prove a theorem. However there is a weird notation that I couldn't understand.
First they present the theorem and then they present two claims. In the first claim it has an arrow that directs to the right, and then a second claim with an arrow that directs to the left. I couldn't understand the meaning of this notation. What does it mean?
Actually I couldn't understand how the two claims prove the theorem and I guess the secret is the the notation. Because the first claim is so strange to be introduced since it's against the goal of the theorem!
The statement that is being proved is a biconditional statement. The key phrase in the theorem is "if and only if". The biconditional statement $A$ if and only if $B$ can be written symbolically as $A\Longleftrightarrow B$.
You prove a biconditional statement by proving $A\Longrightarrow B$ and $A\Longleftarrow B$. That is what the symbols are trying to show you. The first paragraph proves $A\Longrightarrow B$ and the second paragraph proves $A\Longleftarrow B$.