Let x_1 and x_2 be two points of X and define {y_1 = f(x_1)} and {y_2 = f(x_2)}. If \alpha(x_1) = \alpha(x_2) \text{ in } X (i.e. pushing the button \alpha we arrive at the same state whether the initial state was x_1 or x_2) then show that \beta(y_1) = \beta(y_2) \text{ in } Y (the 'same' statement with button \beta on the machine Y^{↻\beta} with regard to its two states y_1 and y_2).

If instead we know that x_2 = \alpha^5(x_1) \text{ in } X (i.e. that starting from state x_1, five pushes of the button \alpha will bring X to the state x_2), show that the 'same' statement is true of the states y_1 and y_2 in Y^{↻\beta}; i.e. y_2 = \beta^5(y_1) \text{ in } Y