Metrics on Space are Lipschitz Equivalent iff Identity Mapping is Lipschitz Equivalence

Theorem
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Let $I_A$ denote the identity mapping on $A$.

Then:
 * $d_1$ and $d_2$ are Lipschitz equivalent


 * $I_A: M_1 \to M_2$ is a Lipschitz equivalence.
 * $I_A: M_1 \to M_2$ is a Lipschitz equivalence.