Sum of CDFs of Binomial and Pascal distribution

Question

I'm trying to prove that for a Binomial RV $x\sim \text{B}(n, p)$ and a Pascal RV $y\sim \text{NB}(k, p)$, their CDF (cumulative distribution function) ($F_X$ and $F_Y$) satisfies: $$ F_X(k-1)+F_Y(n)=1 $$ I know that one can understand the result by considering a sequence of coin-flipping experiment, but I'm curious if one can show it by summing explicitly. I have tried it but there's no progress, so I would like to ask for help. Thank you.