Vector Subspace question $X,Y,Z$ are subspaces of vector space $V$ s.t. $X + Z = Y + Z$, then is $X= Y.$ How to prove it?

If $X \oplus Z= Y\oplus Z$ then is $X = Y.$
No. Let $V=\mathbb{R}^2$, $X=\{\alpha(1,0)\colon \alpha \in \mathbb{R}\}$, $Y=\{\alpha(1,1)\colon \alpha \in \mathbb{R}\}$, and $Z=\{\alpha(0,1)\colon \alpha \in \mathbb{R}\}$. Then $X \oplus Z=Y \oplus Z=\mathbb{R}^2$, but $X \ne Y$.
This is not true. $\mathbb{R}^2 \oplus \mathbb{R}^3 \cong \mathbb{R}^3 \oplus \mathbb{R}^2$ but $\mathbb{R}^2 \neq \mathbb{R}^3$. And there's not just the trivial example of order. In infinite dimensions, it can even be the case that $A \oplus B \cong C \oplus D$ and $A \cong C$ but $B \not \cong D$.