Deducing extra middle term from inequality with two terms

Question

I found the following proof:

I don't understand how does he get the middle term as it seems to appear from nowhere:

$$ ... < \frac{a}{b}\cdot\frac{b}{b + d} + \frac{c}{d} \cdot \frac{d}{b + d} < ... $$

Answer 1

$$\frac{a}{b}\cdot\frac{b}{b+d}+\frac{a}{b}\cdot\frac{d}{b+d}<\frac{a}{b}\cdot\frac{b}{b+d}+\frac{c}{d}\cdot\frac{d}{b+d}$$ is true because by our assumption: $$\frac{a}{b}<\frac{c}{d}.$$