How to Prove Inequality for Equivalent Distances?

[mnb96]

Hello,
two distances $d_1$ and $d_2$ are said to be equivalent if for any two pairs (a,b) and (c,d)

$$d_1(a,b)=d_1(c,d) \Leftrightarrow d_2(a,b)=d_2(c,d)$$

How can I (dis)prove that:

$$d_1(a,b)<d_1(c,d) \Rightarrow d_2(a,b)<d_2(c,d)$$

 

[HallsofIvy]

Use "trichotomy". For any two real numbers, x and y, one and only one of these must apply:
1) x= y
2) x< y
3) y< x.

If $d_1(a, b)< d_1(c, d)$ then it is NOT possible that $d_1(a,b)= d_1(c, d)$ nor that $d_1(c, d)> d_1(a, b)$ which implies the same for $d_2(a, b)$ and $d_2(c, d)$.

 

[mnb96]

There is still something bugging my mind.
I think the part of your proof that is giving me troubles is this:

...which implies the same for $d_2(a, b)$ and $d_2(c, d)$.

How did you prove that "it implies the same for d2(a,b) and d2(c,d)"

Thanks in advance.