Mediant is Between/Corollary 2

Theorem
Let $a, b, c, d \in \R$ be real numbers such that $b > 0, d > 0$.

Let $r = \dfrac a b = \dfrac c d = s$.

Then:
 * $r = \dfrac {a + c} {b + d} = s$

Proof
Suppose $r < s$.

Then from Mediant is Between:


 * $r < \dfrac {a + c} {b + d} < s$

Hence:


 * $r \le \dfrac {a + c} {b + d} \le s$

Hence from the Squeeze Theorem for Functions:


 * $r = s \implies r = \dfrac {a + c} {b + d} = s$