I was trying to know the relationship between four quantities $a, b, c, d$ in order to know when the following inequation is true:
$$ \frac{a-c}{b-d}\lt\frac{a}{b} $$
as far as $b\neq d\neq 0$ of course. So playing around with the inequation we get:
$$ \frac{a-c}{b-d}\lt\frac{a}{b} \quad\iff\quad a-c\lt\frac{a}{b}(b-d) = a-\frac{a}{b}d\quad\iff\quad c\gt\frac{a}{b}d \quad\iff\quad \frac{a}{b}\lt\frac{c}{d} $$
Which means that, if a fraction is less than another fraction, a pairwise substraction yields a lesser value.
Which means that I can apply these substractions iteratively:
$$ \frac ab\lt\frac cd \quad\iff\quad \frac{a-c}{b-d}\lt\frac{a}{b}\lt\frac{c}{d} \quad\iff\quad \frac{a-c-a}{b-d-b}\lt \frac{a-c}{b-d}\lt\frac{a}{b}\lt\frac{c}{d} \quad\iff\quad \frac{-c}{-d}\lt \frac{a-c}{b-d}\lt\frac{a}{b}\lt\frac{c}{d} $$
which is a contradiction because $\frac{-c}{-d} = \frac{c}{d}$.
What I'm not seeying here?