The proof of this lemma is quite straightforward, but we give it for completeness. can be rewritten as

(26) |

with

where . Suppose for contradiction that for some , . We simply permute the two values and , and we show that the new value of after, , is not greater than before permuting, . The difference between and can be easily decomposed using the notation ( ) as the value of (eq. (27)) in , and ( ) as the value of (eq. (27)) in . We also define:

(28) |

We define in the same way . We obtain

(29) |

Proving that can be obtained as follows. First,

We also have :

Here we have use the fact that . This shows that , and ends the proof of Lemma 1.