By definition of Binomial Coefficient: :$\dbinom z w = \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$ Euler's Reflection Formula gives: :$\forall z \notin \Z: \Gamma \left({z}\right) \Gamma \left({1 - z}\right) = \dfrac \pi {\sin \left({\pi z}\right)}$ Thus: {{begin-eqn}} {{eqn | l = \Gamma \left({\zeta - \omega + 1}\right) \Gamma \left({1 - \left({\zeta - \omega + 1}\right)}\right)       | r = \dfrac \pi {\sin \left({\pi \left({\zeta - \omega + 1}\right)}\right)}       | c = Euler's Reflection Formula }} {{eqn | n = 1       | ll= \leadsto       | l = \Gamma \left({\zeta - \omega + 1}\right)       | r = \dfrac \pi {\sin \left({\pi \left({\zeta - \omega + 1}\right)}\right) \Gamma \left({\omega - \zeta}\right)}       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \Gamma \left({\zeta + 1}\right) \Gamma \left({1 - \left({\zeta + 1}\right)}\right)       | r = \dfrac \pi {\sin \left({\pi \left({\zeta + 1}\right)}\right)}       | c = Euler's Reflection Formula }} {{eqn | n = 2       | ll= \leadsto       | l = \Gamma \left({\zeta + 1}\right)       | r = \dfrac \pi {\sin \left({\pi \left({\zeta + 1}\right)}\right) \Gamma \left({- \zeta}\right)}       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = \dbinom z w       | r = \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}       | c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}} }} {{eqn | r = \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right) \left({\sin \left({\pi \left({\zeta - \omega + 1}\right)}\right) \Gamma \left({\omega - \zeta}\right)}\right)} {\Gamma \left({\omega + 1}\right) \pi}       | c = from $(1)$ }} {{eqn | r = \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\pi \left({\sin \left({\pi \left({\zeta - \omega + 1}\right)}\right) \Gamma \left({\omega - \zeta}\right)}\right)} {\left({\sin \left({\pi \left({\zeta + 1}\right)}\right) \Gamma \left({- \zeta}\right)}\right) \Gamma \left({\omega + 1}\right) \pi}       | c = from $(2)$ }} {{eqn | r = \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\sin \left({\pi \left({\zeta - \omega + 1}\right)}\right)} {\sin \left({\pi \left({\zeta + 1}\right)}\right)} \dfrac {\Gamma \left({\omega - \zeta}\right)} {\Gamma \left({- \zeta}\right) \Gamma \left({\omega + 1}\right)}       | c = rearranging }} {{eqn | r = \dfrac {\sin \left({\pi \left({z - w + 1}\right)}\right)} {\sin \left({\pi \left({z + 1}\right)}\right)} \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\Gamma \left({\omega - \zeta}\right)} {\Gamma \left({- \zeta}\right) \Gamma \left({\omega + 1}\right)}       | c = Combination Theorem for Limits of Functions }} {{eqn | r = \dfrac {\sin \left({\pi \left({z - w + 1}\right)}\right)} {\sin \left({\pi \left({z + 1}\right)}\right)} \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\Gamma \left({\left({\omega - \zeta - 1}\right) + 1}\right)} {\Gamma \left({\left({\omega - \zeta - 1}\right) + \omega + 1}\right) \Gamma \left({\omega + 1}\right)}       | c = Combination Theorem for Limits of Functions and rearrangement }} {{eqn | r = \dfrac {\sin \left({\pi \left({z - w + 1}\right)}\right)} {\sin \left({\pi \left({z + 1}\right)}\right)} \dbinom {w - z - 1} w       | c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}} }} {{end-eqn}} Now we have: {{begin-eqn}} {{eqn | l = \sin \left({\pi \left({z - w + 1}\right)}\right)       | r = -\sin \left({-\pi \left({z - w + 1}\right)}\right)       | c = Sine Function is Odd }} {{eqn | r = -\sin \left({\pi \left({-z + w - 1}\right)}\right)       | c =  }} {{eqn | r = -\sin \left({\pi \left({w - z - 1}\right)}\right)       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \sin \left({\pi \left({z + 1}\right)}\right)       | r = \sin \left({\pi z + \pi}\right)       | c =  }} {{eqn | r = -\sin \left({\pi z}\right)       | c = Sine of Angle plus Straight Angle }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \dfrac {\sin \left({\pi \left({z - w + 1}\right)}\right)} {\sin \left({\pi \left({z + 1}\right)}\right)}       | r = \dfrac {-\sin \left({\pi \left({w - z - 1}\right)}\right)} {-\sin \left({\pi z}\right)}       | c =  }} {{eqn | r = \dfrac {\sin \left({\pi \left({w - z - 1}\right)}\right)} {\sin \left({\pi z}\right)}       | c =  }} {{end-eqn}} and the result follows. {{qed}}