(1)

\begin{align*} \sum_{j=0}^{n}\delta_{kj} & =\begin{cases} 0 & n<k\\ 1 & k\leq n \end{cases}\\ & =\begin{cases} 1-\sum_{j=0}^{k-1}\delta_{nj} & n<k\\ 1-\sum_{j=0}^{k-1}\delta_{nj} & k\leq n \end{cases}\\ & =1-\sum_{j=0}^{k-1}\delta_{nj} \end{align*}

(2)

(1)より、
\begin{align*} \sum_{j=0}^{n}f\left(j\right)\delta_{1j} & =f\left(1\right)\sum_{j=0}^{n}\delta_{1j}\\ & =f\left(1\right)\left(1-\sum_{j=0}^{0}\delta_{nj}\right)\\ & =f\left(1\right)\left(1-\delta_{n0}\right) \end{align*} となるので与式は成り立つ。

(2)-2

\begin{align*} \sum_{j=0}^{n}f\left(j\right)\delta_{1j} & =f\left(1\right)\cdot\begin{cases} 0 & n=0\\ 1 & n\in\mathbb{N} \end{cases}\\ & =f\left(1\right)\cdot\begin{cases} 1-\delta_{0n} & n=0\\ 1-\delta_{0n} & n\in\mathbb{N} \end{cases}\\ & =f\left(1\right)\left(1-\delta_{0n}\right) \end{align*}

(3)

(1)より、
\begin{align*} \sum_{j=0}^{n}f\left(j\right)\delta_{kj} & =f\left(k\right)\sum_{j=0}^{n}\delta_{kj}\\ & =f\left(k\right)\left(1-\sum_{j=0}^{k-1}\delta_{nj}\right) \end{align*} となるので与式は成り立つ。

(3)-2

\begin{align*} \sum_{j=0}^{n}f\left(j\right)\delta_{kj} & =f\left(k\right)\sum_{j=0}^{n}\delta_{kj}\\ & =f\left(k\right)\cdot\begin{cases} 0 & n<k\\ 1 & k\leq n \end{cases}\\ & =f\left(k\right)\cdot\begin{cases} 1-\sum_{j=0}^{k-1}\delta_{nj} & n<k\\ 1-\sum_{j=0}^{k-1}\delta_{nj} & k\leq n \end{cases}\\ & =f\left(k\right)\left(1-\sum_{j=0}^{k-1}\delta_{nj}\right) \end{align*}

(4)

\begin{align*} \sum_{j=0}^{n}\delta_{1j}\left(1-\delta_{n0}\right) & =\sum_{j=0}^{n}\delta_{1j}-\sum_{j=0}^{n}\delta_{1j}\delta_{n0}\\ & =\sum_{j=0}^{n}\delta_{1j}-\sum_{j=0}^{0}\delta_{1j}\delta_{n0}\\ & =\sum_{j=0}^{n}\delta_{1j}-\delta_{10}\delta_{n0}\\ & =\sum_{j=0}^{n}\delta_{1j} \end{align*}