The answer is yes, provided that you have access to the inference rule called Hypothetical Syllogism:
If $A$ entails $B$, and $B$ entails $C$, then $A$ entails $C$.
The "Hypothetical" part of the name results from the fact that it is not obvious from the conclusion (i.e., "$A$ entails $C$") that $B$ was involved in deriving it. So in proving that $A$ entails $C$, we may find use for an intermediary, "hypothetical" statement $B$.
So let us see where this Hypothetical Syllogism comes into play in your situation.
From the premise, i.e.:
$\mathsf{append}(\mathsf c(U_1, \mathsf c(V_1, W_1)), X_1, \mathsf c(U_1, \mathsf c(V_1, Y_1)))$
there's only one statement we can infer by using $A$, namely:
$\mathsf{append}(\mathsf c(V_1, W_1), X_1, \mathsf c(V_1, Y_1))$
Now we can apply our given rule $A$ to this expression again, and obtain:
$\mathsf{append}(W_1, X_1, Y_1)$
In conclusion, we have two valid applications of $A$:
$\mathsf{append}(\mathsf c(U_1, \mathsf c(V_1, W_1)), X_1, \mathsf c(U_1, \mathsf c(V_1, Y_1))) \vdash \mathsf{append}(\mathsf c(V_1, W_1), X_1, \mathsf c(V_1, Y_1))$
$\mathsf{append}(\mathsf c(V_1, W_1), X_1, \mathsf c(V_1, Y_1)) \vdash \mathsf{append}(W_1, X_1, Y_1)$
Using the Hypothetical syllogism, we can now infer $B$.