You were probably told, in your earliest riding lessons, that a savvy school horse might hold his breath to expand his chest and make it difficult to tighten the girth---and that's why it's important to recheck your cinch before mounting.
While that advice is good, the reasoning behind it is not quite right. For one thing, a horse isn't likely to voluntarily hold his breath. But even if he did, the first 10 ribs, which lie directly under the saddle, are relatively immobile, which makes it nearly impossible for a horse to expand the diameter of his girth area through lung power alone.
That's not to say that some horses don't "puff up" while they are being saddled. But what is happening is that the horse is tensing his abdominal muscles, an action that expands the width of the chest slightly, allowing the cinch to loosen when he relaxes. The tension may come because the horse is anticipating an unpleasant experience or because the act of tightening the girth causes discomfort, but it's more likely just a natural response to the feeling of having something wrapped around his barrel.
The key to "deflating" a horse's belly is relaxation. First, double-check your tack fit to make sure you are not causing him discomfort when you saddle up. You might need to switch to a
different type of girth, such as one lined with fleece.
When you go to tack up, first place the saddle on your horse's back and attach the cinch loosely, so it is secure but not tight, then finish grooming him or picking his hooves. Before you mount, tighten the girth a bit more and walk him around until he relaxes, then check it again. Remember that your weight will push the saddle down on his back, so you may need to tighten the cinch one final time after you mount.