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The celebrated Assouad embedding theorem has been known for over 40 years. Itstates that for any doubling metric space (with doubling constant $C_0$) thereexists an integer $N$, such that for any $\alpha\in (0.5,1)$ there exists apositive constant $C(\alpha,C_0)$ and an injective function $F:X\to\mathbb{R}^N$ such that \[ \forall x,y\in X \quad C^{-1} d(x,y)^{\alpha} \leq\|F(x)-F(y) \| \leq Cd(x,y)^{\alpha} \]  In the paper we use the remetrization techniques to extend the said theoremto a broad subclass of semimetric spaces. We also present the limitations ofthis extension -- in particular, we prove that in any semimetric space whichsatisfies the claim of the Assouad theorem, the relaxed triangle conditionholds as well, which means that there exists $K\geq 1$ such that \[ \forallx,y,z\in X \quad d(x,z)\leq K\left( d(x,y)+d(y,z)\right). \]

How much can we extend the Assouad embedding theorem?