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The main goal of this paper is to establish a technical result, which provides an algorithm to prove several cardinal inequalities and relative versions of cardinal inequalities related to the well-known Arhangel’skii’s inequality: If $X$ is a $T_2$-space, then $\left|X\right|\le 2^{L\left(X\right)\chi \left(X\right)}$. Moreover, we will show relative versions of three well-known cardinal inequalities.