We can build a heap by repeatedly calling MAX$-$ HEAP$-$INSERT to insert the elements into the heap. Consider the following variation on the BUILD$-$MAX$-$ HEAP procedure:

$\{$

BUILD$-$MAX$-$HEAP $($A$)$

A$.$heap size $1$

for $1-2$ to A$.$ length

PAX$-$HEAP$-$INSERT $($A$,$ A$1])$

$\}$

Do the procedures BUILD$-$MAX$-$HEAP and BUILD$-$ MAX$-$HEAP' always create the same heap when run on the same input array? Prove that they do or provide a counterexample.