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find proof of the theorem

THEOREM 9.2 Let \( f \) and \( g \) be functions defined from a finite set \( S \) into itself, such that \( f \) and \( g \) are their own inverses. Let \( \mathbf{t} \) be a fixed point of \( f \), let \( h \) be the function defined by \( h=f \circ g \), and let \( T \) be the orbit under \( h \) of \( \mathbf{t} \), i.e., \( T=\langle\mathbf{t} ; h\rangle \). Then \( f(T)=T \)

Given that f and g are functions defined from a finite set S into itself, such that f and g are their own inverses, let t be a fixed point of f, let h