Let f$,$ g and h be hash functions that each map binary strings of length $2$n to binary strings of

length n$.$ That is$,$

f : $\{0,1\}2$n $->\{0,1\}$n

g : $\{0,1\}2$n $->\{0,1\}$n

h: $\{0,1\}2$n $->\{0,1\}$n

Suppose that

h$($x$)=$ f$($g$($x$)|$g$($x$)),$

where a$|$b means a concatenated with b$.$ Let CR be the set of all collision resistant hash functions.

Prove that if f in CR and g in CR then h in CR$.$ That is$,$ prove that

$$

f in CR

$$

$$

g in CR

$$ h in CR