Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, mathcal{A})}$ of cardinality ${|A| geq 2}$ and infinite set X of cardinality ${|X| geq | mathcal{A}|}$ , the X-th power ${(A^{X}, mathcal{A}^{X})}$ of the algebra ${(A, mathcal{A})}$ contains a free subset ${mathcal{F} subset A^{X}}$ of cardinality ${|mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${mathcal{I} subset mathcal{P}(X)}$ of cardinality ${|mathcal{I}| = |mathcal{P}(X)|}$ in the Boolean algebra ${mathcal{P}(X)}$ of subsets of an infinite set X.