Suppose we have two finite dimensional random variables $(X_1,\ldots ,X_n),(Y_1,\ldots,Y_n)$. How do we show formally that if $$(X_1,X_2-X_1,\ldots ,X_n-X_{n-1})\stackrel{d}{=} (Y_1,Y_2-Y_1,\ldots ,Y_n-Y_{n-1})$$ then $$(X_1,\ldots ,X_n)\stackrel{d}{=}(Y_1,\ldots,Y_n)$$
Is this even true?
I will leave the explicit solution to you, after giving you this well known fact: Let $X,Y$ be random variables taking values in $\mathbb{R}^n$ such that $X\stackrel{D}{=} Y$. For any measurable function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ it holds that $f(X)\stackrel{D}{=}f(Y).$
To prove this result we notice that for any $A\in \mathbb{B}^m$ we have $$P(f(X)\in A)=P(X\in f^{-1}(A))=P(Y\in f^{-1}(A))=P(f(Y)\in A)$$ Can you prove the result from here?