I've been reading a book on number theory by Hardy and Wright and came across two properties of Farey Series: namely, if $\frac{h}{k},\frac{h'}{k'},\frac{h''}{k''}$ are sequential terms, then the following two theorems are true: $$hk'-h'k=1$$ $$\frac{h'}{k'}=\frac{h''+h}{k''+k}$$
I am attempting to prove that these two are equivalent. The first implies the second is easy to prove, but the other way is tougher. I know it can be done by induction over the Farey Series parameter but I want to attempt it without such. I've tried loads of methods and paths with no clear success. Any hints or advice?