I am sure my post was clear, which cannot be refuted by any deva, mara or sloppy scholar. 

Mindfulness ('sati') means 'to remember'. The breathing as a meditation object cannot be 'remembered'. Instead the breathing is observed, which is called 'anupassi'. 

The purpose of 'mindfulness' in Buddhism is 'to remember' to keep the mind in a state of right view & right intention (refer to MN 117) or 'to remember' to keep the mind free from covetousness & distress (refer to MN 118; MN 10; etc).

Therefore, to practise 'mindfulness **of** breathing' is an impossibility because mindfulness (sati) means to remember rather than to observe (anupassi).  

Instead, the practise is 'mindfulness **with** breathing', which means to remember to keep the mind free in right view free from craving, as instructed in the four noble truths. When this is properly done, awareness/observation of breathing will naturally arise due to the quiet & silence of the mind. 

Thus, mindfulness with breathing is remembering to keep the mind free from craving with awareness of breathing arising as the 'sign' of right mindfulness.