Kindly find below the $5^{th}$ proof of Catalan formula from Wikipedia. After obtaining $2$ formulas for $B_{n+1}-C_{n+1}$, in this case why is it valid to say that $2B_i-C_i={2i+1\choose i}$? Although in general if we have $\sum_{i=0}^{n}x_iC_i=\sum_{i=0}^{n}y_iC_i$, this does not necessarily imply that $x_i=y_i$.
