Normal numbers and selection rules

Given a normal number x=0, x ₁ x ₂ ··· to base 2 and a selection rule S ⊂{0, 1}*=∪ _n=0 ^/t8 {0, 1} ⁿ, we define a subsequence x,=0, {Mathematical expression}·· where {t ₁<t ₂<···}={i; x ₁ x ₂···x _i-1 εS}. x _s is called a proper subsequence of x if lim_i/∞ ti/</t8. A selection rule S is said to preserve normality if for any normal number x such that x _s is a proper subsequence of x, x _s is also a normal number. We prove that if S/∼ _s is a finite set, where ∼ _s is an equivalence relation on {0, 1}* such that ξ ∼ _s η if and only if {ζ; ξζ εS}={ζ; ηζ εS}, then S preserves normality. This is a generalization of the known result in finite automata case, where {0, 1}*/∼ _s is a finite set (Agafonov [1]).