Przestrzeń Y jest proksymalna, jednostajnie sztywna i zawiera

parę DC2.

Dowód. Bezpośrednio z definicji przestrzeni Y wynika, że funkcje wn : R → [0, 1] wykorzystane w konstrukcji są okresowe i dodatkowo blok 12cn występuje relatywnie gęsto w słowie ω⁽ⁿ⁾ = wn(0)wn(1)wn(2) · · · ∈ I dla dowolnego n ∈ N. Wynika stąd, że także w słowie α generującym przestrzeń Y blok 12cn występuje relatywnie gęsto. Rozumowanie analogiczne jak w dowodzie twierdzenia 5.17 dowodzi, że Y jest przestrzenią proksymalną.

Dla dowodu jednostajnej sztywności oraz istnienia pary DC2 wystarczy skorzy-stać z lematów 5.20 oraz 5.21.

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